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THE DERIVE - NEWSLETTER #20 USER GROUP

1. Geometric shapes geckos DUG geckos SWHE Logo DERIVE in frames and gecko DERIVE Please don t forget renewing your member ship for 1996 My wife and I wish you and your family the best for Christmas and a Happy New Year 1996 Best regards met Joe Fiedler at a TI 92 workshop in Houston Unfortunately didn t realize that Joe has been a DUG member so didn t introduce myself Joe you were great enjoyed your and Wade Ellis workshop immensely had many nice hours at the ICTCM including the DUG meeting But this workshop was one of the top events for me Many thanks p 2 E DI TORIA HL The DERIVE NEWSLETTER 1s the Bulle tn of the DERIVE User Group It 1s pub lished at least four times a year with a con tents of 30 pages minimum The goals of the D N L are to enable the exchange of experiences made with DERIVE as well as to create a group to discuss the possibilities of new methodical and didactical manners in teaching mathematics Editor Mag Josef B hm A 3042 W rmla D Lust 1 Austria Phone 43 0 2275 8207 D N L 20 Contributions Please send all contributions to the Editor Non English speakers are encouraged to write their contributions in English to rein force the international touch of the D N L It must be said though that non English articles are very welcome nonetheless Your contributions will be edited but not assessed By submitting articles the author gives his consent for re
2. We would like to rewrite x y 3a 3b 2c under consideration that a 2b 5 Let s reproduce Johann Wiesenbauer s RED function step by step iteration by iteration 1 step Long division and its consequence 3a 3b 2c a 2b 2 3a 6b UY 3a 3b 2c 5 8a 6b 15b 2e 215a 15b 30b 2c remainder 155 2c 2 step We can perform the long division of the polynomials again 15a 15b 305 2c a 2b 215 x 15a 15b 30b 2c 3a 3b 2c 15 54 15b 605b 2c Rem 155 4 60b 2c No more division is possible the next quotient would be zero and the result of the procedure remains the same 75 155 60b 2c The iteration process has come to an end Compare with the result of the RED function from above Things will change if we change the variable order Var1ableOrder b a E 15 a 15 a 8 C 75 RED x y a b 5 a 4 2 4 Who can write a respective function program for the TI s
3. k_ 1 Vn LOG n 11 LOG 1 4nN3 e mV 2n 3 10 5 1 2 PARTS 50 2204226 needs 0 000 sec Josef D N L 20 DERIVE USER FORUM p 7 DISTINCT PARTS n simplifies to the number of decompositions of n into distinct in teger summands without regard to order For example 4 1 3 so DISTINCT PARTS 4 2 DISTINCT PARTS AUX n m IF n lt 2 m 1 1 yY DISTINCT PARTS AUX n k k 1 k_ m FLOOR n 2 DISTINCT PARTS n IF n 1 0 DISTINCT PARTS AUX n 1 1 DISTINCT PARTS 100 444793 needs 16 5 sec The function which is implemented now works also without any auxiliary function Compare the calculation time DISTINCT PARTS n X IF n 16i 1 FLOOR V 8n 16i 1 2 PARTS i 0 eres 274 DISTINCT PARTS 100 444793 needs 0 047 sec All for now Aloha Albert D Rich Applied Logician Albert s letter closes again with his challenge to improve these functions Find now the included file for Harvey Dubner s first of seven consecutive primes separated by 210 That is a new record the previous had been numerous examples involving six consecutive primes in arithmetic progression Albert refers to an article from SCIENCE NEWS vol 148 September 1995 The article starts Searches for patterns among prime numbers have long served as stiff tests of the ingenuity and per severance of mathematicians You can obviously see that not only our column writer Johannes Wiesenbauer is dealing with prime numbers H Dubner
4. ten in the explicit form b than in the implicit one a So we propose the following modifi cation in the ODEI MTH file a We change the name of the functions DSOLVEI GEN and DSOLVEI respectively by SOLVE1 GEN and SOLVEI b After that we add the following functions DSOLVEI p q x y x0 yO a IF inapplicable 2a sz SOLVEI p q x y x0 y0 SOLVEI p q 1 xy xO yO a_ a_ DSOLVEI GEN p q x y c a_ IF inapplicable a SOLVE GEN p q x y C SOLVEI GEN p q 1 x y c a a The meaning of these new functions 1s obvious If the original function DSOLVEI now named SOLVEI gives inapplicable then we try to apply it again writing the differen tial equation in the form b Thus we solve all the examples from above e e DSOLVE1_GEN SEC y TAN x SEC x TAMCGCQD e e 2 2 DSOLVE1_GEN COS y SING Tix IAH y 2 2 2 3 z DSO LUE1 GEM y x 4 x DSOLVE1_GEN 1 x CSC y x ui Z i 1 re LH x y COS y SINCy H Ki 2 A Fa DSOLVE1 GEM y 1 x 2 y 1 x DSDLUE1 GEN SIN y COS y COS x LA x y x 2 y c T 1 Se uve InHix COS y We have suggested to Soft Warehouse to include officially this modification in the ODEI MTH file although some strange examples are still remaining Ex 7 xy y x Ddx x y 2xy x 2y 2x 2 dy 0 taking from 1 p 354 ex 31 which admits t
5. SOLUTIONS v 5 1 VARTABLES v_ APPEND SOLUTIONS v_ CVARIABLES v_ SOLUTIONS v L 1 VARIABLES CvV 2 33 Vo V 1 Some details of the 1995 version for DERIVE 3 have been changed to make the file suitable for DERIVE 6 10 Then produce the following VIETADEM DMO demo file You can do it with any text editor or using the Edit gt Annotation facility of the recent DERIVE versions More details can be found in DNL 4 The comments are the lines with the leading semi colon The must be entered as Annotation without the semi colon together with the respective DERIVE commands This initializes the random number generator Press ENTER RANDOM 0 Try to solve the equations Continue with ENTER task VIE 2 x Congratulations if you are right Press ENTER for the next bundle SOLU task D N L 20 Jan Vermeylen amp Josef Bohm Vieta at Random p55 Check your solutions pressing ENTER task VIE 5 a Let s finish this sequence with a package of 10 SOLU task ENTER will present all solutions task VIETA 10 It more than two answers are wrong go on practising You can then run the DEMO file again SOLU task Interrupt the demo and check the solution using the SOLVE command VIETA 3 VIETA 5 After checking the results you may run the DEMO again VIETA 10 If you are doing in with DERIVE 6 you might face problems saving the file as a demo file with extensions dmo only I d
6. Sina so that the three solutions found in succession using M ller s Method can be compared with these exact solutions As a more realistic non trivial example the reader could try the problem on page 192 of the DERIVE manual for version 2 60 e z 0 after viewing M ller s Method applied to x 1000 0 on the program following The advantages of M ller s Method over the classical Newton Raphson Method as employed in the manual would then be apparent The brief Muller program that follows could be made even more compact but there are advan tages in a more lengthy display in terms of converting formulas in Muller s Method literature to a DERIVE program 1 Precision Approximate PrecisionDigits 15 2 Notation Decimal NotationDigits 10 3 F x z ee ag amo uomo CN 4 1 2 3 4 F x2 F x1 F x1 FCx0 F x2 F x1 5 ca iz cb _ Ca i cb x2 xl xl x0 x2 xl F x1 FCx0 xl x0 z gt 2 2 6 ma IF c2 c2 4 F x2 c G2 de cr we gro 2 Area e e ee An 7 ba LE nn 2 Fix ma T8 3 9 Fix 1000 x 10 MULLERC 0 5 0 0 5 0 12 xl xz x3 p D 5 a 0 5 s L 0 5 4000 S 0 5 4000 SIE 352 d 4000 0505943852 LI 1774703 3 O 80894382 1 1774703 1 5253355 4 L 1774703 1 5253355 17 43907 11 1 5253355 17 439307 B 2551095 6 17
7. cos x xsin x c2 Example 10 de ceive esis b 2 xy 2 65 c5 l _g 2 GF c7 Example 11 de solvely y iy E x 2 c9 c9 1 waded x t2 c9 xtc9 1 J y S X T2 C9 x c9 21 and xtc920 cT0 c Example 12 E T T deSolvely y 1 x y Hnlcoslerer er 3er tc12 As there is the same CAS implemented it is no surprise that NSpire and the handhelds are behaving pretty the same Examples 9 10 17 and 18 E ssbralcatelotherPrantolciean ue ee ssbralcatelotherPrantolciean ue des5olveu a x Sini X H pe J oos x sini SSC HSC a deSolvel y ran tufan a x ul uly 3 4 6 637 _ i x ESS l j be it a deSolvel y Ze e a deSoluelu i u UI Irun x y e le 21 Zye a zc pd 5 4 g9 ee PC epo yiz y Indy 3 u 4 839 4 desolyvetyt y xty 2 a 4g 0luetu u 3xl1In v x vu RAG AUTO x at Male RAD AUTO FUN rzi Male FUNC zh z p26 Comments on the Differential Equations D N L 20 This is the rest of the list done with TI NspireCAS Example 13 T TN 2 desolvelii i y 2 0 well ll tetrix salteti Example 14 deSolvely y 1 y seg y ecsa e15 c15 see c16 Example 15 desolvelyy y xy ylinly c17 1 x 618 Example 16 desolvelly 2 ut E elle 19 x c20 Example 17 it eae EY eee ds b t b 0 x EE b 2 E c21 x c22 Example 18 de salve ye b in S A c24 x Example 19 de slvelasisi
8. se d 2c The reader is friendly invited to double check the solution Example 24 v y and y 1 v 0 di cd ENER dy d ee Integrate y l C l v Iny Ind Inc y p28 Comments on the Differential Equations D N L 20 2 C y Solve for v y l v d Ae y 7 pe EE E NC I take the positive root and separate the variables y yY dy y dx Integrate again Jc y Jc y x k Solve for y y 2 tfc x k asked MATHEMATICA to do the job Ini DSolve y x v x 1 v x 2 0 v x x Out 2 MIO y x ent pg c l M AN ert ci i Fortunately my result from above was confirmed The next contribution causes problems George Douros wrote his package for DERIVE 3 and it does not run with DERIVE 6 solve solutions and incompatibilities with the then used characters found George Douros email address in the Internet and asked if there is a re cent version available received an answer the following morning Hello Josef Boehm am glad to hear from you and sorry that cannot be of help had written a much more powerful ODE package at the same time with the Special Functions pack age for Derive 6 Some novelties in the simplification machine of Derive I then had a few arguments with Albert Rich prevented my new package from becoming functional have since the end of 2003 stopped working on the package Feel free to d
9. v w 1v l LN v 1 c y 2 ATAN Cv k ed Ex 20 type b d Q4y 104 y k 6 each c y x ty Jk Ex 21 type b y 9y INC 9 y Kk 3 y 2 Selec x 2 c y 2 x Sy Jk c 3 DSOLVE2 O 9 0 IK 3 X cl e ce As the illustration shows the differential equation in the next example can also be solved taking into account that it is a linear equation The next example corresponds to a not incomplete linear equation although y does not appear Ex 22 y y cos2x 2sin2x DSOLVE2I w y COS 2 x 2 SIN 2 x inapplicable The two final examples show some limitations In the first the expression of the ob tained solution seems not to be very satisfying In the second taking from 2 p 261 we do not obtain a solution In both cases the problem 1s caused by the resolution of the first order equations Le from DSOLVEI GEN m Ex 23 type d l y yy 2 1 DSOLVE2I 1 v yw a a ne esse k 2 2 k IF y 0 e y e 1 g ie k IF y gt 0 e J y D N L 20 Comments on the Differential Equations p21 Ex 24 type d yy 1l y 0 References 1 Llorens Fuster J L Aplicaciones de DERIVE Analisis Matematico I Calculo Servicio de Public de la Universidad Polit cnica de Valencia 1993 2 Soft Warehouse DERIVE User Manual Version 27 Honolulu 1994 Some comments for the revised version
10. y t Bildet man noch den Betrag p x t voll so kann man mit DERIVE die Kurve in Polarform 5x 9 o zeichnen Wendet man dieses Verfahren nun auf die oben hergeleitete Parameterdarstellung der Kardiode E 2 cosa cos2a zC sin O sin 20 an erh lt man nach mehreren Umformungen eine Glei chung 4 Grades mit der Variablen cos a die DERIVE nicht l sen kann Verschiebt man allerdings die gesamte Kurve so dass ihre Spitze in den Koordinatenursprung fallt E 2 cosa cos2a Ge Q sin sin ail und schlagt den oben angedeuteten Weg ein erh lt sing sing man tang Um den Radius f r die Polarform zu erhalten kann man also bei dieser Coso COSA speziellen Lage der Kurve direkt den Betrag der urspriinglichen Parameterdarstellung verwenden und erhalt wieder nach einigen Umformungen mit DERIVE die einfache Polardarstellung der Kardiode ID p Q cosa cos2a SS Q sin sin 2a 2 1 cosa Next we will find the polar form of this curve We need for each point P t e curve its dis tance p from the origin and the angle o see the sketch above If there exists a reversible unique relation between t and o then t can be presented as a function of t t which leads to x t y t o or x 9 y o with e aretan M or tano A x t x t If it is possible then we solve this equation for t and substitute t for t With p x t q y t o we are able to plot this cur
11. 1 b e Sal v tan lease Loge c25 8 71 c26 ory peas atan c25 e 1 c25 e Example 20 1 e nu e e f r Example 21 deSolve de Solvely 9 y xy y c30 e 7X 4 29 7 Example 22 _ SECH 2 sinl2 x 5 deSolvely y cos 2 x 2 sin 2x x y c31 e 4 c32 e F Example 23 de Solveli y 2 1 1 Ta a or dy xt es4 or ih ELLE m Hess 33 y 21 EN c33 21 I change the settings from Real or Complex Format gt Real to gt Rectangular 12 u 2 2 desolveli ly y y x y l Loan 1 J039 l f 39 1 039 dy x c34 Example 24 de Eu ap 1 b 0 m bhi y cd1 bl IERT dy xtc4d2 or dy x c42 D N L 20 Comments on the Differential Equations p27 You can see that the TIs are well prepared to solve 2 order ODEs of this kind which are appearing in Llorens Fuster s selection felt challenged to check my freshly acquired and remembered knowledge in solving 2 order ODEs applying my skills on examples 23 and 24 Example 23 ley y y voy and mp Y Ev ug EE dx dy dy wd S B Integrate y 1 v iny Ine e Inc y ed ev p c c v Solve for v y d ly Se v I take the positive root and separate the variables C dy INT dir Integrate again C J 2 y de In y e y 2 x k Solve for y x k 2x 2k eNe ale Ie ee SE SE el Lee l y Y Mm Substitute c gt 2 C 2 poene 2 get
12. 2 SUBST dy y ATAN y C 3 COSCy 2 2 X y y X 1 3 DSOLVE1_GEN L 2 2 XV 2 X y X 2 y 2 X 2 2 2 LN x 2 x 2 y 1 4 ATANCy 2 It is interesting that the Tl family shows no problem to solve this rare kind of differen tial equation without hesitating ernennen ul de5aolus u 39 CC et D EE xong Init 2 x 2 z Iniyut 1 z tandiu m a deSolvel y x2 y 2wytRi 2y 2 h inl i tani ga ln M t 2 AXHA JE GQxXu 2 u 2 x 1 5 x u AIN H RAD AUTO FUNC 2 730 J L Fuster Improvements on the Resolution of ODEs Finally we show an example that is neither a limitation of this function nor in general of the program The linear differential equation Ex 8 y ycosx sin2x 1 p 366 ex 56 1s not correctly solved if we don t select Manage Trigonometry Expand DSOLVE1_GENCSIN 2 x y COS x 1 SINHCx SIHECx SIN Z x dx y amp Trigonometry Expand SINCx B 2 SIN x This is no problem for DERIVE 6 E fiser d Prantofeiesn ue a eh y cos x sin 2 x x S But it a problem for the voyage 200 and ae sinc ster a e5iNOO axe ae TI NspireCAS as well which can be re Enge EE cos x ut u 22 sinix COSH solved expanding the trig expression before deSolve cosCO u 4 uy 2 sintx costx gt P integrating the differential equation y mio SINK 2 sin x 1 d
13. APPEND v ZCi p 12 CYCLE p CREATE p 1 DD 13 CYCLE p zi CIE 27 04033 130 9 578 53 2 28 AD 548 cag 28 dis bI 28 ER 301741 21 1E e E 16073403 22 737 098 70D E E e E EC 23 17 34 IB 36 44 4B 50 51 ZB 38 70 521 Two nice graphics from Belgium ios 2 G P Speck the author of the contribution Mueller s Method lives in Wanganui New Zealand The Wanganui River is the main river on the Northern Island of New Zealand One of the famous places in the Wanganui region is the Bridge to Nowhere Regards to wonderful New Zealand p10 J L Fuster Improvements on the Resolution of ODEs D N L 20 SOME IMPROVEMENTS ON THE RESOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Jos Luis Llorens Fuster Universidad Polit cnica de Valencia Departamento de Matem tica Aplicada 46071 VALENCIA SPAIN First order equations The general solution of the first order differential equation P x y dx q x y dy 0 a can be obtained using function DSOLVE1 GEN p q x y c which is incorporated in the ODEI MTH file see 1 chap 5 p 333 2 chap 9 6 p 252 always that this equation be e Separable e Linear e Homogeneous e Exact e Equation having an integrating factor which depends only on x or only on y When the differential equation is not one of the previous type 1f we simplify the men tioned function we obtain inapplicable If we want to obtain the particular solution
14. H 10 10 10 5 10 10 10 DH 10 10 10 10 References 1 Samuel D Conte and Carl deBoor Elementary Numerical Analysis An Algorithmic Approach McGraw Hill Kogakusha Ltd Auckland 1980 pp 120 127 2 Steven Schonefeld Numerical Analysis via DERIVE MathWare Urbana 1994 pp 53 70 Some ideas for the revised Version of this DNL 1 GP Speck writes about a program for demonstrating and performing M ller s Method In 1995 it was a huge progress to have a list of functions calling each other working in the sense of a program Now we can write real programs as a whole I ll collect all procedures in one program without loosing the insight into the single steps Application of ITERATES has an advantage in demonstrating the iterative nature of the process but the disadvantage that its syntax is sometimes difficult for students to understand A sim ple loop might be easier to follow This realized in my program version D N L 20 Additional Comments on M ller s Method p43 m meth u v n ca cb c c ma xd dis table P rog tables xd ee a eee a a xps og nod Loop Ifn n RETURN table cats CSUBSTCu x vga SUBSTCu x vi1Z lJ 0v42 vy2 1 cb SUBSTCu x v42 SUBSTfu x vi1lli v42 vull cis fea cbi v43 wl CZ Ca te lvy3 v2 dis ss J c2 2 4 SUBSTCu x vy3 c ma IF ABS c2 dis gt ABS c2 dis c2 dis c2 dis xd w3 2 5SUBSTlu x vySi ma vis v42 vis x4
15. Solving a Univariate Equation G P Speck Wanganui New Zealand A short background information Numerical Analysis with DERIVE St Schonefeld p 53 70 Muller s method is an extension of the secant method The secant method approximates F x by a straight line first degree polynomial in order to find an approxi mation to a root of a function With M ller s method F x is approximated by a quadratic polynomial Josef M ller s Method see the Reference at the end of this note for solving univariate equations has some very significant advantages over many numerical methods available In particular the often troublesome problem of finding a guess reasonably close to a solution sought especially in the complex number case is not a critical issue in using Muller s Method One possible objection to the method is that convergence can be relatively slow compared to some other numerical methods how ever in many problems where this s an issue Muller s Method can be used with a small number of iterations to produce quite adequate initial guesses for solutions which can be found using one of the fast methods for which reasonable initial guesses are required In the DERIVE program for M ller s Method which follows the example x 1000 0 is given as an illustration and solved in detail to show exactly how the program is used The exact solutions to x 1000 0 are easily computed to be 10 5 Sina and 5
16. USER FORUM D N L 20 across a concept or topic which can be investigated from an educational point of view and which has in their opinion the potential to contribute to our understanding of how students learn mathematics via a computer Regards Martin Lindsay e mail Martin Lindsay vut edu au FAX 6136884050 DNL I sent some own materials to Martin and I hope that there are some other DUG members who will support his work Are there any suggestions for projects It could be interesting to compare pro jects done in different countries und under various circumstances and conditions Hopely we will hear about results Additionally I can recommend the Resources for Calculus Collection from the MAA which I bought in Houston last week and transferts in French You will find more information in the Book Shelf In the last DNL you could find an interesting request about a square root simplification I received two respective answers one from the Fachhochschule Osnabr ck and another one from Albert Rich SWHH which explains the issue FHS Osnabruck The fact described by Mr Propper in DNL 19 has more odd aspects If one having defined the interval 1 1 for x tries to simplify expression 1 he will obtain the expected result 5 Only expression 2 derived by DERIVE will not simplify Although DERIVE recognizes 1 and 3 as equal they are treated differently when they are simplified The next example shows the same the declarati
17. and H Nelson ended up using seven com puters running continuously for about 2 weeks to find the sequence Now they are thinking about going to eight consecutive primes They estimate that it would take 20 times longer at least 2 5 com puter years to accomplish this search on their souped up personal computers Josef Harvey Dubner s first of 7 consecutive primes separated by 210 prelog9523545124T7T0095109075790519922997722909030492995315919539 52 131059061742150447508967213141717495151 The following verifies the difference of the 7 consecutive primes and that DERIVE s primality is working good v ITERATES NEXT PRIME n n p 8 You might simplify 5 to see the 8 huge prime numbers VECTOR v SUB n_ 1 v SUB n yn DIMENSION v 1 210 210 210 210 210 219 129 32 p 8 Benno Grabinger Playing Cards Shuffling D N L 20 Playing Cards Shuffling with DERIVE Benno Grabinger Neustadt Germany In DNL 12 Mr Chuan from Taiwan posed the following question Is it a surprise that after perfectly shuffling only 8 times a deck of 52 cards the origi nal position of the cards is restored With the help of DERIVE it is no problem to understand what is going on The perfect shuffling is described by the permutation 12 3 4 49 50 51 32 1127 2 28 25 51 26 52 It is well known that each permutation is a product of cycles having no common elements The func tion CYCLE p creates a product of cycles which represents t
18. as 1st x 0 or simply 1st x For example to find the general solution of xy x xy x 2 y x x if y x is a known homogeneous solution enter GENERAL x y2tx yl y x x 0 6 Conclusion ODE MTH is more a mathematics than a programming package I am a mathematician not a programmer I have tested it against similar packages in other symbolic mathematics programs and found it competitive I will not make any further claims on its speed and efficiency and leave its eva luation to those who will use it This not simple modesty it is the fear and dread of the easy counter example to any of my claims I have been faced with many of those in writing this package With all due respect to the power given by the programming functions in the symbolic mathematics programs I have been taught once more that a mathematical problem can be solved only by mathematics and not by programming functions The latter are assistants and Derive s A Mathematical Assistant par excel lence The hints given here on how ODE MTH works are obviously not enough to help a user ex pand improve or fully exploit it in a Differential Equations course but they have made this a long article ODE MTH together with its accompanying files is available via the Derive Internet Mailing List See the Additional Resources appendix of the Derive Manual Office Home Prof George Douros George Douros T E I of Larissa Kolokotroni 3 Larissa 41110 Lariss
19. continues by claiming that ef I Again we could calculate ef coso cos2o cos4g cos8g cos3go cos5 cos 69 cos 79 l l STE e e D N L 20 Johann Wiesenbauer Titbits 6 p57 TN and using z as well as 16 Y hes k l after a lot of tedious computations we could finally arrive at the desired result Try it out by yourself Is there a way to make DERIVE do these calculations for us You won t be taken by surprise if I tell you there is actually one The following utility function can be used quite generally to reduce a poly nomial expression u in one variable modulo an equation v which amounts to adding the rule v to the other rules for the handling of polynomials RED u v ITERATECRHS v QUOTIENT Cu LHS v REMAINDER u_ LHS v u_ U Bu 2 By setting u e f and taking for v 3 5 6 7 10 1012 14 Z Z HZ Z Z 2 2 2 we get fe 2 16 k_ REDIe f 2 z 1 1 Koi Therefore e and fare both roots of the quadratic equation x e f xtefax EE which has the solutions Now we have come across a small problem namely is Qcxsv fex or vice versa Again DERIVE can be of help in showing that the first alternative is valid APPROXC lim COS p COS 2 9 COS 4 COS 8 9 0 780776 po2 n 17 Do you see the achievement The polynomial equation of degree 16 has been broken up into two polynomial equations with fewer terms It takes no Gau to ha
20. das bevorstehende Weihnachtsfest und ein gl ckliches Neues Jahr 1996 zu w nschen Herzliche Gr e LETTER OF THE EDITOR pl Dear DUG Members Let me start with a congratulation It is your merit that we can celebrate the 5th year of the existence of the DUG As we have now a member from Namibia who joined us two months ago we can really say that the DUG Is represented on all five continents We will express our warm welcome For 1996 I am looking forward to having an interesting enrichment in the DNL Bert Waits form the Ohio State University OSU and Bernhard Kutzler SWHE two TI 92 special ists suggested including a TI 92 column in the DNL They both have promised to submit the first contributions and to answer TI 92 related requests Some contributions of this issue focus on differential equations ld like to ask our friends who are not involved so much in these applications for patience can promise for 1996 a really sportive DNL a student s pro ject on ski jumping geometry of the soccer ball base ball a comparison of sportsmen s performances and hopefully an article DERIVE and the tennis net Among the files you can find a small Christ mas gift There are some stereograms in the folder STEREO Try your magic eyes by loading them into any graphics program e g IRFANVIEW You can either print the pic tures or inspect them on the PC screen used POPOUT LITE to create the pictures
21. dy amp dx c row c y does not appear using the change v y we transform it to a first order differential equation in v v x The general solution of this equation is obviously another first order equation d x does not appear the changes y v y v v transform it to a first order differential equation with the variables v 1n place of y and y in place of x The general solution of this equation is again another first order equation It is well known that in the ODE2 MTH file are only two functions to solve the dif ferential equations presented in a and d In particular AUTONOMOUS r v We apply it to solve equation d written in the form y r y v with y 2v DERIVE simplifies it solving for v after applying the previous change AUTONOMOUS CONSERVATIV E r x y Xxo yo Vo We apply it to solve equation d written in the form y r y where the initial conditions y x V X v are sup posed So it cannot be used to obtain the general solution of the equation The following shows what Jos Luis proposed in 1995 Then we will have a look how the utility file SecondOrderODES mth is supporting solving second order DEs in our times J L Fuster Improvements on the Resolution of ODEs For case a Le y u y 2 u v it is very easy to define the function FALTAYX u v x y C k x INT 1 u v c y INT v u v k Similarly for the incomplete equations of kind b i e y 2 u
22. erstellen Wie man sofort sieht gilt z d d 2 x y rS COS X11 C2 r SIM x 1 COSCX11 2 z 2 2 x y 4 r C0Sfe 1 Zur Elimination von o aus dieser Gleichung l st man noch die in cos quadratische Gleichung x rcos 1 cos a nach cos a auf und setzt ein DERIVE liefert bei diesem Vorgehen die algebrai sche Kurvengleichung der Kardiode 2 2 2 2 2 o x y Jee oo derex y J der ey Die entsprechende Polarform lautet p r C1 COStp We found all representation forms for the Cardioid You may follow the calculation in the re spective DERIVE file Thomas Weth A Lexicon of Curves 7 Calculating the length of the curve and the EE KI Kc Prantolciesn uel enclosed area gives interesting results An E aimee F a r sintay 1 costo zu interpretation of these results can be found 2 sintod cost 1 r investigating the cardiod as a special epi cycloid fa Eco ico as 16 0000 r 12 5 zei cost 43 las ogee QJRCZXrXxC l cosiQp252 2 0 H Zem MAIN FAD AUTO FUNC E 20 Merkw rdige Ergebnisse ergeben sich beim Berechnen der L nge der Kurve und des von ihr einge schlossenen Fl cheninhalts Mit dem CAS berechnet man IECH y a da 16r Damit ist die Lange der Kurve ein ganzzahliges Vielfaches des zu ihrer Erzeugung verwendeten Kreises Iris l l A gt f play da 6r z also wieder ein ganzzahliges Vielfaches der Fl che des Kreises Eine voll st ndige Int
23. factor u x y x 7 ODE x 2 y 34x l y 2 y1 8 LN y Cl 3 2 Equations of degree higher than one in y x For equations of degree higher than one in y x ODE tries to solve for one of the variables x y y and proceed from there The equation vw 3xy y has the solution 9 ODE y 2 y1 2 3 x yl y og a eo one ee De er er p32 George Douros Differential Equations with Derive D N L 20 or equivalently 3 2 2 3 file sel ex oc we Cl 0 Gem 2 od which was actually obtained by choosing to solve the original equation for x this 1s simpler than solv ing for y or y This can be seen by issuing the command 212 26 SOL VAB E y Lay Note that ODE also appends the singular solution when it can be found to the primitive In this case 9x 4y 0 is the singular solution of y y 3xy y The solution of higher degree equations is usually found in parametric form ODE does not automatically attempt to eliminate the parameter because in its present form it may lose parts of the solution The user can in such cases use ELIMINATE to simplify the solution 12 ODE 2y Lizy y Iny 13 XS y y x y 0 14 ELIMINATE 113 o 2 2 y 4 cl 3 2 A cl 4 Equations of the 2 Order With second order equations of some complexity ODE starts by assuming that the equation is given in an unnatural form a simple DE has undergone transformations in both the dependent and
24. ihres Aussehens unter dem Namen Kardioide oder Herzkurve vom griechischen kap ta bekannt ist Dieselbe Kurve l sst sich auch als helle Linie in einer mit Tee gef llten Tasse beobachten Genau genommen handelt es sich bei diesen Kurven um halbe Nephroiden Ozanam erwahnt die Kardioide in seinem ma thematischen W rterbuch Dictionnaire mathe matique Amsterdam 1691 die heute noch g lti ge Namensgebung geht allerdings auf Castillon zur ck De curva cardioide 1741 Die Kardioi de ist mit der fraktalen Geometrie zu neuer Aktualitat und Popularitat gelangt sie bildet in grober N herung die Umrandung der Ikone der Chaostheoretiker n mlich der Mandelbrotmenge volkst mlich Apfelmannchen gt If you put a cylindrical ring on a plane table then the falling in sun rays will be reflected and they form approximately a focal curve which is called cardiod or heart curve One can observe this curve in a cup filled with tee or coffee Actually the car diod consists of two half nephriods In a rough approximation this curve forms the border of the pn chaos theorists icon the Mandelbrot set Konstruktion der Kurve und Herleitung ihrer Gleichung Construction of the curve and derivation of its equation 3 2 una Eine genaue Konstruktion ergibt sich mit fol gender Vorschrift Konstruktion der Kardioide als katakaustische Linie als Einh llende einer Geradenschar G
25. like to remind you that you have to save it as vietadmo dmo including the quotes 2 Is it a feature or is it a bug x Lion 72 2 The function works but if by chance we have two solu Sg tions with z z then DERIVE automatically solves the VIETA S z equation instead of only expanding the product of the two respective polynomials 2 wz 7 t l t 28 DERIVE 5 amp 6 allow programming So we can collect the single procedures in a compact program which excludes the special cases z z2 and z or z 0 7 list_ zl z2 Ca z1 a z2 b z1 b z2 d z1 d z2 e z1 de zz Det ed ee zen ze z1 i z2 Ck z1 k z2 o z1 o z2 p z1 p z2 t ZU IE 22 ZI 7 2 1Y ZN 2 Tee 2291 VIETA_ n 1 Prog qnd CES Loop 8 Lr 1 m RETURN tbl Dp pe xs If p l py2 0 pyl py2 z 0 tbl APPEND tbl EXPAND Clist pil p 2 CRANDOM 15 1 0 12 1 tot DJs p56 Johann Wiesenbauer Titbits 6 D N L 20 Titbits from Algebra and Number Theory 6 by Johann Wiesenbauer Vienna This time I would like to deal with a purely algebraic problem to make up for the fact that we have concerned mostly with number theoretic issues so far And what could be more algebraic at least from the historical point of view than the solution of polynomial equations As you all know DERIVE can easily cope with polynomial equations in one variabl
26. of this differential equation satisfying the initial condition xo yo we can use the DSOLVEI p 9 X y X0 yo function Thus the general solution of the equation taking from 1 p 353 ex 3 Ex 1 y sin x 2 cos y can be obtained simplifying the expression DSOLVE1_ GEN cos y sin x The general solu tion will be obtained depending on the constant c e IR In the ODEI MTH file which is now FirstOrderODEs mth Josef there are some in dependent functions for these types of differential equations SEPARABLE GEN p q x y c for the separable differential equation having the form y p x q y LINEARI GEN p q x y c for the linear DE having the form y p x y q x HOMOGENEOUS GEN tr x y c for the homogeneous EXACT GEN p q x y c for the exact J L Fuster Improvements on the Resolution of ODEs INTEGRATING FACTOR GEN p q x y c for these equations having an inte grating factor which depends only on x or only on y In the last three functions we suppose that the differential equation is written in the implicit form a The DSOLVE1 GEN func tion chains the previous functions using the corresponding tests to identify each type of the equation For example if the differential equation satisfies op eq Oy x then it is an exact one Thus the definition of the function contains a sentence of the type IF DIF p y DIF q x EXACT GEN p q x y c From this we can conclude that the efficie
27. solution From 4 v tan x c and substituting in 3 2 jo ESCORT E ee 2 2 cos x c But can also try solving the DE directly by integrating twice BU iunii a Integrate dx l y x c tan v gt v y tan x c dy tan x o Integrate Fr xc g y In cos x c k Let s have a type b example Ex 21 don t recognize the DE as a differential equation with constant coefficients and apply again an appropriate substitution for reducing the order of the equation dv vdv v dv dy 5 dy v y and y 9y 9y k dx Integrate dx dy dy dx l 9y k 9ydy vdv Integrate In GES mu 9y vi X m tc k 3 2 D D D D D D D ee ee oy k m am trying to bring the solution in its explicit form D N L 20 Comments on the Differential Equations p23 J9y k 3y e Ou k sfer 3yy Spr BUET 9 3x 3c Rx 28 2 3x 3 3x43c x 3c k e 6 6 As you can see now we obtained finally the solution in the form how it is given by solving the DE as ahomogeneous DE with constant coefficients using the characteristic equation 6y e e e 3x 3x y ce c e Of course one can do all the symbolic manipulation supported by CAS too But sometimes it s nice to do it manually isn t it We miss an example of type d so let me choose Ex 16 H yry d d l v y and y EE gt v y u Separation of vari
28. the authors and a DUG member since long sent one copy 170 pages It is an interesting collection of work sheets for students with accompanying support material for teachers Merci Dominique Dominique has submitted a contribution which is one of my favourites for the next issue 7 Matem ticas con DERIVE en la economia y la empresa Alfonso Gonzales Pareja ra ma ISBN 84 7897 201 3 Exchange for DERIVE Teaching B rse f r DERIVE Unterrichts materials in the DNL materialien im DNL The wheel has not to be invented twice Das Rad muss nicht zweimal erfunden werden I can offer Binomial Theorem GCD amp LCM System of Coordinates in English and in Ger man as well Modelling Word Problems with DERIVE DERIVE Days Leeds Spring School on Teaching and Learning Mathematics with DERIVE and the TI 92 Trinity and All Saints College Leeds 13 15 April 1996 For receiving an announcement please send a self addressed A4 envelope to John Monaghan Centre for Studies in Science and Mathematics Education University of Leeds Leeds LS2 9JT D N L 20 Liebe DUG Mitglieder Gleich zu Beginn mochte ich uns allen zum funf j hrigen Bestehen der DUG gratulieren Mit dem ersten Mitglied aus Namibia das ich recht herz lich in unserem aller Namen begr e sind nun alle f nf Kontinente in der DUG vertreten F r das Jahr 1996 steht eine interessante Auswei tung des DNL in Aussicht Bert Waits von der Ohio State Universi
29. 18 15 28 20 0 70245 742 2 788 7403 10 i 0 70346742 6 5525512 10 et 0 70346742 2 00 8468 10 i 18 28 20 20 0 70245742 6 5525512 10 i 0 70346742 2 00 8468 10 i 0 70346742 3 9241405 10 CG 20 3 m meth x 10 5 i 10 D 10 20 21 22 2 2084165 0 34420843 2 208465 D 34420843 c 2 2084153 0 34420843 15 2 204153 0 34420843 2 2054153 D 34420843 0c 2 208465 0 34420843 0 20 2 wanted to illustrate the process by deriving the quadratic functions and plotting them 24 25 20 Tl 28 29 30 31 32 33 34 together with the newly found approximation d qx ax bx c H fix 1000 x Initial guesses are 4 2 5 see page 421 CSOLUTIONS g 4 f 4 a gl f 2 a q 5 f 5 a b p These are the coefficients of the parabola 3 18 1040 You may use the FIT function to obtain the parabola too 4 f 4 bs FIT Lx ax bx c_ Pes ae 3 x 14 x 1040 5 FS 2 SOLUTIONS 3 x 18 x 1040 D x 15 85912687 21 85912687 Which zero of the approximating parabola is the appropriate one 5 15 85912587 5 21 85912687 10 85912686 26 85912686 Ye take the solution with the smaller distance to x3 which is 15 858 The next parabola should pass the points 2 f 2 5 f 5 and 15 858 fi 15 85 S T 2 pec FIT Lx a x Bex c_ 5 FES 15 85917687 F61
30. 4 x cos ocos 49 C x ax4 20 2 We let DERIVE put the finishing touches D N L 20 Johann Wiesenbauer Titbits 6 p59 17 DEE J17 1 17 17 NAI 1 las _ es 32 32 8 8 32 32 8 8 d C sorurzons d X HU 2 JC 4 38 4 17 170 3 17 17 JC34 2 Pro S Ur 1 EE n a 8 16 16 16 JC 4 38 17 170 3 417 17 G4 2 417 J17 1 8 16 16 16 0 9324722294 0 09226835946 APPROX lim COS p 0 932472 p22 n 17 That s it There is only one thing that is slightly disturbing namely that DERIVE didn t return the term for cos o in its most beautiful form which is according to Gau l l coso SE EUM 2417 ail kl 25 344 17 But isn t this a bit of a tall order in view of this huge formula Far more important is the fact that both forms of cos show that apart from the basic operations only square roots are used for its calculation From this follows easily the stunning fact that the regular polygon with 17 edges can be constructed by using compass and straight edge only cf 2 for the details As Gau was able to show this in his Disqusitiones Arithmeticae 1801 this 1s possible for a regular polygon with n edges if and only if n is of the form n 2 Lan r20 8520 where fi f f are pairwise distinct Fermat primes Thus we ve returned happily to number theory where we will continue next time References 1 WuBing H Mathematisches Tagebuch 1796 181
31. 4 von Carl Friedrich Gau Akedemische Ver lagsgesellschaft Leipzig 1981 2 Strubecker K Carl Friedrich Gau Princeps Mathematicorum Bild der Wissenschaft 5 1977 118 126 Three additional comments from the editor The file TITIBITS6 MTH is among the accompanying files and it contains the DERIVE functions and the complete calculation p60 Johann Wiesenbauer Titbits 6 D N L 20 gt Some time ago a DUG member sent a DERIVE file calculating the 17 roots of the equation He refers to an article written by Friedrich Freytag DdM 3 1992 pp 188 213 I recom mend reading this article There one can find a description how to construct the side of a regular 17 edge I intend incuding this construction in DNL 21 2 Johann Wiesenbauer has invented a remarkable function RED u v We compute a poly nomial expression under consideration of an additional condition which is in my opinion a kind of special substitution Johannes told me that he had thought that this would be an advantage of other CAS packages and he wanted to make this possible with DERIVE too Three easy to follow examples for applying RED u v ee up c E e 2 REDCx y 3 x a b 102 3 2 E 18 b 3 b 54 c b 4 c 123 c 180 c c 3 c 20 i xus deg pp YCD Fr 1 RED 3 6 y a B a 2 P 18 b 9 b cu l IT Il 1 2 3 8 c 3 RED 3 0 vi j nt 2 4 e RED Pw Boa e228 5 E BD 9x5
32. 43907 5 251055 9 6881509 7 SW 2561065 9 6881500 9 987120 amp 95 6881509 9 9871208 10 000023 9S 9 947120 10 000023 LI 10 10 000023 10 10 alae 10 10 LI l2 12 z1 10 3 1000 x mrs CPs xc ae 14 MULLER 0 5 0 0 5 0 5 xl x2 0 5 0 L 0 5 5 15 0 5 5 amp 660254 2 5 5 6 660254 5 amp 660254 5 5 amp 660254 5 amp 660254 5 5 8 660254 5 8 660254 5 Elo z2 iz 5 amp 8 5602b54 3 1000 x 17 Fix x zll x zz es EE a MULLER mO nJ APPENDC xl x2 x3 pl ITERATESCu m m ni x3 0 5 amp 660264 8 660254 i amp 660264 8 660254 i amp 660254 p42 G P Speck Muller e Method D N L 20 18 MULLER 0 5 0 0 5 0 5 xl xz x3 p 0 5 0 0 5 0 D 0 5 5 8 660253999 1 18 0 5 5 amp 550253899 5 8 560254037 2 2 5 amp 600253999 5 amp 66025405 2 5 8 560254037 tr 3 5 amp 565602B54037 5 8 6602540s 2 5 58 660254037 4 r Ln 5 amp 6602540s 2 5 amp 66025405 2 5 8660254037 i 20 zl 10 z2 5 8 660254 2 3 5 8 650254 i 3 21 Fix 1000 x 22 MULLERC C 4 2 5 0 10 xl xf x3 p E z 5 0 2 15 85812587 1 5 15 855 758 8 260222125 2 15 559126587 8 266222135 9 934452212 3 9 206222135 359 9544527 12 23S 955D 48758 4 i D 9344522712 9 999074829 10 00000014 5 D 999074829 10 00000014 10 D 1 0 00000014 10 10
33. 5 85912687 d 22 4591 2686 x l21 013888D x 441 408 7315 2 SOLUTIONS 22 85912586 x 121 013888 x 841 4087313 x 3 972324047 9 266222136 15 85912687 3 972324047 15 85912687 9 266222136 19 83145081 6 592904733 D N L 20 Additional Comments on M ller s Method p45 This ts the third parabola 5 f 5 T 35 FIT x ax bx ec 15 85812687 f 15 85912687 9 266222136 f 8 2656222136 z 36 30 12534900 x 272 5808374 x 265 2290375 2 37 SOLUTIONS 30 125248 x 272 580B8374 x 265 2290376 x 8 934452211 0 8862271639 The next figure shows the function together with the first three parabolas Ax 1000 x p46 Additional Comments on M ller s Method D N L 20 The undefined messages starting in row 14 are caused by the restricted accuracy The values of x2 and x3 are so close that their difference is internally rounded to zero and then the division in column F gives an undefined result All other error messages are the conse quence However we have result in the last complete row 13 Document Settings General Settings recommend the Document Settings as en shown on the right Exponential Format Normal Real or Complex Format Rectangular 4uto or Approximate Auto Vector Format Rectangular Base Decimal v v v v v v v v Unit System SI Apply to System Reset t
34. D vec coef zero vecl 10 HURWITZ ROwW base 1 nj i VECTORCELEMEHMT Base n 2 7 97 13 j 1 n 11 hurwitz_matrix i WECTORCHURWITZ_ROW base 7 nj 1 1 ni 12 hurwitz VECTORCPOSITIVECDET MAIM MINOR Churwitz_matr x k1JJ k 1 n n 13 testis A ELEMENTChurwitz 1 1 1 14 stability ts Ifltest n asymptotically stable solution asymptotically unstable solution declare the coefficient matrix of the system 4 1 l5 ae 5 0 The characteristic polynomial of the matris 1s H 16 Pix al x 4ex 5 This is the vector of coefficients of the polynomial 17 vec coef 5 4 1 The Hurwitz matrix of the polynomial 4 5 15 Hurwitz matrix O 1 Vector List of the determinants of the main diagonal minors Its elements are equal to one if the determinant is posite and equal to zero if it is not positive 19 Hurwitz 1 1 Now will test if the trivial solution is asymptotically stable 20 stability asymptotically stable solution Another example a sb 21 ais 1 1 22 Px vec coef hurwitz matrix hurwitz d 2 Z Esse du dex 2 fey Ee NR ee 0 1 24 stability asymptotically unstable solution And finally a third example 1 2 0 25 ac 3 Z2 O 3 1 26 Px vec coef hurwitz matrix hurwi tz T52 13 0 d E SET x p excep 125 5557 See See a 0 EL 28 stability asymptotically stable solution p40 G P Speck Muller e Method D N L 20 M ller s Method for
35. Hannes and others this is a real challenge I have put these functions and the next ones together in one file NUMBEXT MTH You can merge this file to NUMBER MTH and so improve your NUMBER MTH Josef Albert Rich 2 Dear Josef Enclosed is an article that appeared in a recent Science News about consecutive prime numbers that you might find interesting thought that these seven 97 digit primes would make a good test of DERIVE s NEXT PRIME function The enclosed printout shows that DERIVE does correctly recognize primes of this size Since my October 16 1995 letter we have simplified and renamed the function for computing the partitions of a number This is how my comment in this letter should read PARTS n simplifies to the number of decompositions of n into integer summands without regard to order For example 4 1 3 2 2 1 1 2 1 1 1 41 sg PARTS 4 5 PARTS AUX n m IF n lt 2m 1 1 PARTS AUX n k k k m FLOOR n 2 PARTS n IF n 1 0 PARTS AUX n 1 PARTS 4 5 PARTS 50 22042260 needs 8 05 sec Recent Derive versions have another PARTS function implemented It looks very bulky but it is much more efficient than the earlier one and it works without the auxiliary function PARTS n IF n 2 1 FLOOR APPROX X 1 mN k 2 X IF GCD h k 1 COS T X i k 1 2 MOD i h k 1 2 i 1 k 1 2nh k 0 h 1 k 2N6e n 24n 1 6k e mV 24n 1 3k 6k mV 24n 1 6k mNY 24n 1 k 24n 1 3 2
36. I d like to show how DERIVE 6 and the Tis are treating the ODEs of 2 order First of all I ll solve one or the other of the presented differential equations in the traditional way This might support the students understanding of the implemented CAS procedures Let s start with Ex 9 type c p l 1 y y xsinx X We substitute y v and y v in order to reduce the given DE to a linear DE l 2 y v x sinx x The standard technique for solving linear DEs of the form y p x y 2 q x is to find an inte grating factor p x and then multiply both sides of the equation by this factor lg l ER Sg ae HI seen x l 1 v v sinx x x Lo rl 3 c sinx Integrate wrt x x v cosx k 2 v xcosx kx gt y xcosx kx y xsin x cosx TC The last step is easy work We can integrate directly applying integration by parts p22 Comments on the Differential Equations D N L 20 proceed with Ex 12 type a 1 yay found in my old textbooks that substituting v for y might be successful So let me try perform the substitution in 1 and integrate the 17 order DE by separation of variables vdv _ 2 09 ee ao Integrate dy y l e In 1 Jor 2 dv dv 1 y dx Integrate 4 dx liy x tan vic This is the parameter representation which is given as result of example 9 Finally we can try to have an explicit form of the
37. THE DERIVE NEWSLETTER 20 ISSN 1990 7079 THE BULLETIN OF THE USER GROUP Contents Letter of the Editor Editorial Preview DERIVE User Forum Benno Grabinger Playing Cards Shuffling with DERIVE Jose Luis Llorens Fuster Improvements on the Resolution of ODEs George Douros Differential Equations with DERIVE Ales Kozubik Asymptotically stable solutions G P Speck M ller s Method Thomas Weth A Lexicon of Curves 7 The Cardiod Jan Vermeylen amp Josef B hm Vieta at Random Johannes Wiesenbauer Titbits 6 The regular 17 Edge revised Version 2009 December 1995 D N L 20 INFORMATION Book Shelf D N L 20 1 DERIVE Projekte im Unterricht MU Jahrgang 41 Heft 4 95 Friedrich Verlag 30917 Seelze 2 Materialien zum M Unterricht mit Computer und DERIVE Landesmedienzentrum in Rheinland Pfalz 1995 Hofstr 257 56077 Koblenz 3 Learning Linear Algebra through DERIVE Brian Denton Prentice Hall 1995 ISBN 013 122664 9 351 pages 4 Business Calculus today with Spreadsheets and DERIVE R L Richardson Saunders College Publishing 1996 ISBN 0 03 017554 2 416 pages 5 Resources by Discovery MAA Notes 27 31 The Mathematical Association of America 6 transferts Cahiers de la Cellule Recherche Innovations P dagogiques Numero 7 Derive Utilisation d un logiciel de calcul formel C R I P Rectorat de l acad mie de Lille 20 rue Saint Jacques 59000 Lille France Dominique Lymer one of
38. a 41223 GREECE GREECE Tel 041 611 061 72 Tel 041 234 866 Fax 041 610 803 D n 1420 A Kozubik Asymptotically stable Solutions p37 Asymptotically stable Solutions of the Systems of Ordinary Differential Equations A Kozubik Bratislava Slovakia A given system of ordinary differential equations of the first order can be a mathematical model for a number of mechanic biologic economic etc systems The behaviour of this system is described by the solution of the respective system For most of the systems we require that their be haviour will be similar to the behaviour of some given system This requirement is described exactly by the stability of the solution Any system y g ty 1 can be reduced by the transformation y x v where v is any solution of 1 into the system x f t x f t o o 2 where f t x 2 t x v g t v System 2 has the trivial solution which is the transform of the solution v of system 1 So we can deal with the stability of the trivial solution of system 2 The trivial solution of system 2 is said to be stable iff for any t and any e there exists t e such that for all initial values amp satisfying c and for all t 2 t the solution u t t of the initial problem x f t x x satisfies the inequality u t t amp lt E The trivial solution of system 2 is said to be uniformly stable if number 6 does not depend on the initial point t M
39. ables dx dy dy dy dv Integrate y v Inv Inv Ink gt y k v k 2 Separate variables again ex SE Integrate k y x Inc In 7 y X x S yac et ecke would leave it to the students to compare this solution with the solution given on page 19 as the result of Llorens Fuster s function You can also double check the solution by substituting into the given DE must say that enjoyed comparing the traditional way resolving the equation s and their outcomes with the CAS results How does DERIVE 6 perform DSEOLVE2 DFILT IR FO L5e 2 simplifies t an explicit gen eral solution of the linear second order ordinary dif ferential equation Qe SE a ae a ous T in terms of arbitrary constants cl and c2 Note that the last two arguments can be omitted if they are vari ables and you are satisfied with the names cl and c2 If no method applies or the equation cannot be con verted to an equivalent one having a p and a q that are independent of x DSOLVE2 returns the word inapplica ble Online Help p24 Comments on the Differential Equations D N L 20 Example 8 1 1 DSOLVE2 0 x SIH x inapplicable x 1 v 2 SE y x SIN x 1 x osco qu J X X 3 deer x S E 7 v c x x 05 x x CX 4 DSOLVEI_GEN c x x COS x 1 x y k CO0S x x SIN x y sch 2 As you can see DSOLVE2 does not apply one has to perform the reduction
40. e up to degree 4 Therefore to be a real challenge for DERIVE the degree of the polynomial in question has to be greater than 4 What about the polynomial equation z 1 0 which was solved by Gau when he was only 19 years old Actually Gau started his famous diary on March 30 1796 with the following entry Principia quibus innititur section circuli ac divisibili tas eiusdem geometrica in septemdecim partes etc cf 1 Of course the application of the built in SOLVE to this equation is definitely out of question But what we could do is to follow the thoughts of genius Gau and enjoy their originality and elegance leaving all the drudgery to DERIVE Are you ready for it Then here we go To begin with it suffices to determine the solution i l Ac z e cosg ising with pum of the equation above since all other solutions are merely powers of it In the outline of the solution given by Gau in his letter to Gerling he actually determined cos o which amounts to the same thing To do so he first introduces the notations a cos cos 4o b cos29 cos Ro c cos 3o cos 20 d cos 6o cos Jo as well as e a b and f c d Then he states that according to a wellkown theorem the equality l e f 2 2 holds Using _ ly e bs 17 k coskp z Z 6 z this can be derived without DERIVE in the following way 2 ee l z 1 1 e f Ycosko Y z f H 2 22 2 z l 2 Gauf
41. ee factors The fact that the first order differential equation obtained after doing the substitution of the variables belongs to these kind of equations which are recognized by DSOLVEI GEN The possibility of solving for v in the solution of this equation The fact that the new differential equation obtained in this process is again of a type of equation which is recognized by DSOLVEI GEN J L Fuster Improvements on the Resolution of ODEs In the following functions we study the way how DERIVE identifies the kind of n complete equation and as a consequence applies the right function to solve it TESTX u v IF d u x 0 FALTAX u v inapplicable inapplicable TESTY u v IF 3 u y 0 FALTAY u v TESTX u v TESTX u v TESTV u v IF 3 u v 0 AND 3 u x 0 FALTAXV u v TESTY u v TESTY u v ODE2I u v IF d u x 0 AND 9 u y 0 FALTAYX u TESTV u v TESTV u v DSOLVE2I u v w ODEZ2I SOLUTIONS u w 1 1 v We apply the ODE2I function for the second order equation y u x y v The result is inapplicable f the equation does not belong to any of the types a b c or d In other case we obtain the solution with the restrictions noted previously for the kinds c and d We apply the DSOLVE2I function for the second order equation u w v x y 0 where w y Thus is the main function where it is not necessary solving w y before Given an equation it
42. ehen von einem Punkt auf dem Umfang eines Kreises Strahlen aus die an der Kreislinie re flektiert werden so umh llen die reflektierten Strahlen die Kardioide This is the instruction to find the cardiod as a catacaustic line A point on the perimeter of a circle is the initial point of rays which are reflected at the perimeter The family of the reflected rays form the cardiod as their envelope see the sketch Thomas Weth A Lexicon of Curves 7 Now in 2009 we can produce the locus of the reflected segments on the graph screen of the Tis working with the Cabri application Tl Nspire offers the same occasion D m E GG h F F U Ge ri P LI FT LI F d SU L I Lt SR SW MIA A RES ox ln 7 1 ei The right graph shows the cardiod on the TI NSpire s Graphs amp Geometry Screen We proceed with Thomas Weth s text from 1995 Zunachst soll nun eine kartesische Koordinatendarstellung der Kurvenpunkte hergeleitet werden Dabei ist zu beachten dass die Kurvenpunkte genau die Ber hrpunkte mit den Kurventangenten sind P sei der Punkt von dem die Strahlen ausgehen In obiger Figur wird der Strahl in Q reflektiert Dann gilt f r die Steigung des Strahls QT m tan Au erdem gilt nach dem Sinussatz im Dreieck OQT und unter Ber cksichtigung des Vorzeichens f r P sin rsin den Achsenabschnitt n der Geraden QT also n 2 E cx dv 0 30
43. erly resolve the problem will teach DERIVE that factoring the difference of two squares is easy to do DERIVE is an obedient student but it has to be taught each and every detail On another subject your readers may be interested in the following recent additions to the utility fle NUMBER MTH The function CONTINUED FRACTION u n approximates to a vector of n 1 partial quotients of the continued fraction of u For example CONTINUED FRACTION fe 8 approximates to 2 1 2 1 1 4 1 1 6 If question marks appear in the result use the Options Precision command to increase the precision CONTINUED FRACTION is defined as follows CONTINUED FRACTION u n FLOOR ITERATES 1 MOD x X u n The function PARTITIONS n simplifies to the number of decompositions of n integer summands without regard to order For example 4 1 3 2 2 1 1 2 1 1 1 1 so PARTITIONS 4 5 The following definition of PARTITIONS was contributed by James FitzSimons p 6 DERIVE USER FORUM D N L 20 PARTITIONS AUX Nym mi IF m 1 1 SUM PARTITIONS AUX n k m 1 k k mn FLOOR n m PARTITIONS n SUM PARTITIONS AUX n m 1 m 1 n PARTITIONS 4 4 Unfortunately for n greater than about 50 the above functions take too long because of explosive fan out on the recursive calls to PARTITIONS AUX Are there any DUG mem bers who can come up with an efficient definition for PARTITIONS that Soft Warehouse can include in NUMBER MTH Aloha Are there
44. erpretation dieser Ergebnisse erhalten wir wenn wir die Kardiode als eine spezielle Epi zykloide eine Kreisrollkurve betrachten Anhang Appendix Definition von H llkurven Definition of envelopes _ F xyja Aa NF x a INS N a Aa y e Aerch A Sei F x y o eine Funktion so dass F x y a 0 f r jedes o eine Kurve beschreibt F x y o bestimmt dann eine Kurvenschar mit dem Scharparameter a Im obi gen Fall sind die F x y o die algebraischen Gleichun gen von Geraden Die Koordinaten der Schnittpunkte P zweier benachbarter Kurven erf llen dann die Glei chung F x y a F x y a 0 und speziell auch F x y a Aa F x y a 0 Aa l a OF x y d a RT F r Aa 0 geht die Gleichung ber in 0 Erh lt man nun durch Elimination von o a OF x y aus den Gleichungen F x y 0 und 0 wieder die Gleichung f x y 0 einer Kurve a so nennt man diese die H llkurve der Kurvenschar F x y o If F x y a is a family of curves with parameter a then we can find the envelope of these curves by eliminating the parameter from the two equations OF X y a Oa F x y a 0 and 0 If this procedure gives an equation f x y of a curve then this curve is the envelope p54 Jan Vermeylen amp Josef B hm Vieta at Random D N L 20 I had the intention to place here a short Utility provided by Sergey Biryukov As I have the strong impression that the contents o
45. esolyvetanstl1 gt x uM NC ESI Male RAD AUTO 5 DSOLVELI GEN SIN 2 x y COS x 1 Mode Settings SIN x Eh e 2 SINCx Vy 2 E Input Simplification Output Transformation Direction Exponential auto Trigonometry Auto This is the recent form of DSOLVE1_GEN how it is implemented in DERIVE 6 Jos Luis suggestion for improving the version from DERIVE 3 was obviously accepted by Soft Ware house and extended to include other cases too DSOLVE1_GEN p q x y C a If inapplicable a_ INTEGRATING FACTOR GEN p q x y C If inapplicable a HOMOGENEOUS_GEN p q X y C If inapplicable a SEP GEN p q x y c GEN HOM GEN Em XE p16 J L Fuster Improvements on the Resolution of ODEs D N L 20 Second order equations When a second order differential equation y u x y y is incomplete i e when some of its terms in x y or y do not appear sometimes it is possible to solve it directly or reduce it to two first order equations So a yandx do not appear the equations are of the form y f y The parametric equa tions of the general solution as a function of the parameter v and the constants c and k dv vdv x c y k f v f v b y andx do not appear the equations are of the form y f y The general solution are can be written in one of the following expressions as a function of the constants c and k ve y 42 fo
46. f this issue has become heavy enough I will do this in DNL 21 In stead of this I reprint an international product Belgian amp Austrian When I was in Belgium Jan Ver meylen gave me a lot of work sheets and DERIVE files Among them were some very interesting at tempts to use the random number generator for creating exercise examples These files were be pa tient you will find them in one of the next year s DNL poly_frac mth to practise factoring polynomials rekrij mth to solve problems with arithmetic series quadr_eq mth to create various quadratics At the DERIVE conference in Honolulu our working group had the idea to produce a How it could look work sheet for quadratics and factoring polynomials We decided that I should add a demo file in combination with some exercises I remembered Jan s quadr eq mth and this is the result First load VIETAUTI MTH as Utility file 1 zl RANDOM 21 10 z2 RANDOM 31 15 z3 RANDOM 11 5 2 list Ca z1 a z2 b z1 b z2 d z1 d z2 e z1 Ce z2 f z1 f z2 a z1 g z2 Ch z1 Ch 22 i zl i z2 k z1 k z2 o z1 o z2 p zl p 22 t zl t 22 x zZl x 2 y zDI y 22 zD G S il VIE n va Kee lin x zl x z23 D n TE zi 3 X AN VIETA n an SEE e ni i qc SC 4 RANDOM 15 1 SOLU v VECTORCIFCDIMCSOLUTIONS v_ VARIABLES v 2
47. he integrating factor u y e TI How ever in this way the general solution is not obtained The new definition of the DSOLVEI function does not act because the given result is not inapplicable DERIVE identifies cor rectly the type of differential equation but is not able to obtain the solution because it cannot evaluate the integrals involved p14 2 DS LUE1 GEM x y E 4 2 ez 2 ATAMCY E ty e 1 amp fe RK 2 Z xy DSOLVE1 GEM E x y 2 x y LH x Z x e ATANCY Z J L Fuster Improvements on the Resolution of ODEs D N L 20 2 2 1 x y 2 x y x 2 f 2 ATAN CJ Zorte v Du 13 2 J y dy 2 3 Z Y x Z E y X 1 FA ex 2y Z x Z 1 g 2 iy As it is shown in the third expression in the previous illustration it 1s solved if we write the equation in form b This is due to the fact that now the integrating factor is 2 y 1 3 ytl which is leading to an exact differential without appearing the mentioned inte grals Obviously this example is very rare To solve this problem we need to modify more deeply the ODEI MTH file As you can see even DERIVE 6 is not perfect in solving ODEs It needs the same rewriting as in 1995 2 2 2 2 1 DSOLVE1_GEN x y y x 1 Xx y 2 X Y X 2 y 2 x 2 2 2 2 y 2 ATANCy x y 1 2 e 2 e x y 1 2 SUBST dy y ATANCy 2 COS y EA e SINCy
48. he permutation p Applying this function to the given permutation p leads to 1 27 14 33 17 9 5 3 2 28 40 46 49 25 13 7 4 29 15 8 30 41 21 11 6 31 16 34 43 22 37 19 10 32 42 47 24 38 45 23 12 35 18 36 44 48 50 51 26 39 20 52 It can easily be seen that this product consists of 6 cycles of length 8 and one cycle of length 2 There fore if p has been applied 8 times the original distribution will be restored 1 CYCLE MTH by Benno Grabinger 1995 2 RER 3 i VECTIORIK k 1 n 1 n 1 4 v VECTORI IFIMOD i 2 0 een 2 2 2 5 p u v pou 3 4 hu Ge ege ge Tess 1 ee Er Des 17 ea es 1 Ig 2B 7L dl l1 27 2 268 3 79 4 3 5 31 6 32 y 31 8 34 9 35 10 36 11L 22 23 24 25 26 2F 7B 28 30 31 32 33 34 35 36 37 38 38 40 41 37 17 3B 3133 39 14 40 15 41 1G 47 I1 43 18 M 9 45 20 45 Al 42 43 44 45 46 47 48 49 50 51 52 47 22 48 23 49 24 50 25 51 26 52 7 VALUE Ci p ELEMENTCELEMENT p 2 i 8 Z s p DELETE ELEMENTCITERATES VALUECj Di J 5 1 EQUAL v w If DIMENSION w 0 0 9 If ELEMENT v 1 ELEMENT w 1 1 EQUAL v DELETE_ELEMENT w 1 INO 23 If DIMENSION v 0 0 10 If EQUALCELEMENT v 1 z 1 1 INCDELETE ELEMENT v 1 z Benno Grabinger Playing Cards Shuffling CREATE p i v E ioe DE WE D s E V 11 Sab Ierd ER bp I CREATE p i 1 v CREATE p i 1
49. ied to look exactly like 24 29 x eReal 0 k sReal 0 2 30 ODE 28 k k 2 5 31 y X aco e1 X F21 k k Ped UE F21 2 0 x J 2 2 Si d 4 3 Searching for Symmetries The other major method used to bring an equation to its natural form is the search for symme tries in the dependent and independent variables Consider for example the readily integrable equa tion 32 where Q t is an arbitrary function If 32 is subjected to the transformations 33 37 the resulting equation 38 1s in what we called an unnatural form ODE can be impressive in solving it only because it realizes that 38 is invariant under the transformations x v gt a x fy and automatically reverses 33 37 to recover the original natural form 32 y2 Q yl 33 TRANSFORM X y2 Q y1 LN x 2 34 O yl x 4 y2 x Elsa 35 TRANSFORM Y 34 LN y 36 Vie x 2 y2 yl Vix a x on E y 2 y George Douros Differential Equations with Derive p35 37 NUMERATOR FACTOR 36 Trivial 2 2 Vds oos Zu NEIL ya ey ol Y 39 ODE 138 1 40 lim au x c1_ IEN lim Dee ae c2_ u gt f x R u E gt LENNKS 5 Additional Utilities The package makes internal use of some utilities that the end user may find useful These utili ties are e ELIMINATE uo ol tries to eliminate o from a parametric set v v 2 Coe eae To sim plify for example the solution of the eq
50. independent variables resulting in the given complicated equation A search is initiated for the reverse transformations that will recover the original simple DE Three major methods are used changing the independent variable x reducing the equation to its canonical form and scaling transformations sym metries in the variables x y y y 4 1 Changing the Independent Variable x Asking ODE to solve the equation sin x y sin x cos x y 4y 20 we get 16 equ 1 SIN x 2 y2 SIN x COS x yl 4 y 17 ODE equ 1 20 H8 y ei elle c_ san 2 uv 2an D N L 20 George Douros Differential Equations with Derive p33 How did ODE proceed When the solution algorithm reached the internal function CHANGE X equ 1 was transformed by letting x gt 2arctane computed from the coefficients of y and y 19 INVERSE INT N DIF equ 1 y DIF equ 1 y2 x x x 2 20 2 ATAN E 21 TRANSFORM X equ 1 420 222 4 y y2 which is a simple equation to solve 4 2 Reducing an Equation to its Canonical Form The canonical form y EE 2pq t 4pr z y 0 of the differential equation equl py qy ry 0 is obtained by letting y ye where 5 fade It can easily be shown that equation equ2 pif y q f y r f y 0 has exactly the same canonical form as equ and that its solution is a multiple of the solution of equl This fact is used by ODE to solve equations like equ2 by reducing them to their cano
51. ional Comments on M ller e Method p47 Finally I d like to visualize the single steps of the algorithm by plotting the approximating pa rabolas supported by a slider in the Graphs amp Geometry Application For this purpose calcu late the parabolas passing the three points using my polreg program which is part of my per sonal statistics library called statistik see DNL 2 000000 15 00000 15 8591 1 a 9 934452 0 999074 10 0000 29 ETA TT Ki dE eio olo alan olo Hi Mi jlele SSES ze besche SSC 290 9 50 HERR See eer ee eran statistik polreg 2 cldle Se ERI a A m Bo e h a We F a Wd be introduce a slider and visualize the approximating parabolas step by step This works pretty well until the accuracy of the system has reached its limit f2 parabs k E f2 x parabs k fei H x f x NN 05 IEE f2 x parabs k INN i f2lx parabsik 5 EE 0 703467422 This is a 2 example the equation mentioned by GP Speck on page 40 p48 Thomas Weth A Lexicon of Curves 7 D N L 20 Ebene Algebraische und Transzendente Kurven 7 Thomas Weth Wurzburg Germany Die Kardiode The Cardiod Legt man einen zylindrischen Ring auf eine ebene Tischflache so werden einfallende Sonnenstrah len am Inneren des Rings reflektiert Die reflek tierten Strahlen h llen n herungsweise eine Brennkurve ein die wegen
52. is therefore strongly recommended that the command Manage Branch Any this is actually the default DERIVE INI setting be issued before attempting to solve most ODEs which contain symbolic parameters Exponents of symbolic powers x of the independent variable x appearing in the coefficients of y y and y in linear 2 order ODEs must be declared either positive or negative since ODE needs to test the behaviour of the equation at its singular points 2 1 Solutions in terms of Special Functions Many 2 order ODEs will lead to solutions involving Special Mathematical Functions These functions Bessel Kummer Hypergeometric are defined as arbitrary in the package and will not sim plify They can be simplified by loading the additional package FUN MTH after ODE MTH Loading both ODE and FUN in one session will use up most of the available system memory in a plain 640K Derive session In such cases the user can save the solutions returned by ODE and simplify them by loading FUN MTH in a new session No such problems occur with DeriveXM 2 2 Local Variables and Functions I know of no way to introduce local variables or functions in a Derive package I have there fore used its ability to understand almost all ASCII characters in order to imitate protected local variables and functions This is done in the file ODE ASC where all variables are translated to ASCII characters above 180 The default variables x y
53. is sufficient to substitute w y v y as one can see in the following examples taking from 1 p 357 Ex 9 type c y xsinx T each ee EN x d Ex 10 type c y y x 4 y DSOLVE2I v w x w k k BD 0 d lt lax lt 0 ee lass 0 WEN A A e r Jose Luis Fuster presents another result So m D SR bb l d like to double check the solution define y fi ae EIE e a function y x and substitute in example mE 4 4 10 2 This solution seems to be correct y GO y GO x y 0 20 J L Fuster Improvements on the Resolution of ODEs Ex 11 type c y y 2 0 4 y 2 1 2 DSOLVE2I v w 1 v d d 2 IN J v 1 v v4 1 x c Ju 1 y d Ex 12 type a y y zl A IN v 1 DSOLVE T w v k 2 Ex 13 type c 1 x y xy 220 DSOLVE2I l x w x v 2 DI A A A IN Q x 1 x kN 1 x yz Ex 14 type a y 41 y 0 2 DSOLVE21 w JL v p c ASIN V y J l v Ex 15 type d vy SP DSOLVE2I y w v Ex 16 type d y ey 2 DSOLVE2T v vw Ex 17 type d y y yy 0 3 3 y 2 DSOLVE2 T w v vw 2 kW 2 key zc 3 Ex 18 type d y y Iny DSOLVE21 w v LNCy J L Fuster Improvements on the Resolution of ODEs Ex 19 type a 2yy x14 y A DSOLVEZI 2
54. ncy of this function depends not only on the previous five definitions but also on the corresponding test On the other hand it is possible to modify the way how to present the differential equation a For example we can write it as dy p x y b dx gay So in order to solve it we simplify the expression DSOLVEI GEN p x y q x y 1 and obviously we expect to obtain the same solution or an equivalent one However the following illustration shows the behaviour of DERIVE v 3 01 with respect to the separable differential equation of example 1 2 2 DSOLUE1 GEMCCUSCu SIN x J inapplicable Pa COS CQ DSOLVE1_GEN 1 2 alMEx COT x Tally c 1 2 SEPARABLE_GEN COS fy 2 SIMCx TANCy c COTOGO Obviously the bug is not in the resolution of the equation because as you can see the SEPARABLE GEN function works correctly but in the fest But this 1s not an excep tional example the following differential equations taking from 1 p 353 ex 15 18 43 45 and 46 are also of separable form Ex 2 x y 2 y x 4 Ex 3 x y 2y 1 2 y 1 x Ex 4 sec xtan y dy sec y tan x dx 0 Ex 5 x y y Z 12 x escy Ex 6 sin y cos y dx cos x dy 0 p12 J L Fuster Improvements on the Resolution of ODEs D N L 20 2 3 DSOLUE1 GEM y x 4 x inapplicable DSULUE1 GEM y 1 x 2 y 1
55. nical form and proceed from that point by pattern matching with canonical forms of equations with known solutions Equation hyp ode is a simple case of a hypergeometric differential equation and is directly solved by ODE 1225 AYP e E ER E EE y 23 ODE hyp 5 1 3 1 24 y x ol_ Vx F21 x c2_ 21 2 0 x 2 2 2 2 y X Suppose we transform hyp by first letting x gt x and then yo The resulting equa tion looks awesome This is what we previously characterized as an unnatural form By reducing this equation to its canonical form however ODE can solve it in a natural way as if only by making sub stitutions in 24 25 k 2 TRANSFORM X hyp x k 2 k k 2 k 1202 27 u ee Se eye ee ee ae ae Kk ee EFT 27 Q x 3 TRANSFORM Y 26 y Q x p34 George Douros Differential Equations with Derive D N L 20 2 k 2 28 2x Les Q x y2 k q d xo 0 2x 2x 0 x a x 4x 0 x 2 5k a x ER dx dx kr 2 d 42 2 rd 2 d 2 2 2s E atx Q x 2x EE eeh Xx dx dx dx 2 d 42 2rd 2 d 2 2 2X 26 Q x 4x EE x E RE E E EH P3k Q X dx dx dx Before solving the above equation k must be declared positive or negative because ODE needs to test the behaviour of the equation at its singular points and Derive is reluctant to perform certain simplifications without this assumption for k We also optionally declare x to be positive so that the solution 31 be fully simplif
56. o Defaults The entries for the cells are C1 b1 G2 f d1 f c1 d1 c1 D1 b2 H2 f2 g2 e1 c1 E1 b3 I2 f2 h2 e1 d1 C2 d1 J N i2 2 4 h2 f e1 D2 el K2 when abs i2 j2 gt abs i2 j2 i2 j2 i2 j2 F2 fle1 f d1 e1 d1 E2 ze1 2 f e1 k2 try to perform the next step according to G P Speck s DERIVE routine in order to find the next complex root recommend deleting some rows in the table before changing the function and or the initial guesses The expressions in the cells might become too bulky and the system might hang up See below the complete entry of cell D6 in grey It is better to start with less rows and then proceed by copying down row for row Don t forget to save between the steps d 1000 x Done x 10 A BH DW B By EEE DEE qx 0 500000000000 0 500000000000 0 EK 0 0 0 500000000000 4 9999999999999 8 66025 10 50 9 5t W 050 0 500000000000 4 9999999999999 8 6 5 8 6602540378443 i 5 500 10 4 4 9999999999999 8 5 8 6602540378443 i 5 8 6602540378443 3 6602 5 5t 5 5 8 6602540378443 i 5 8 6602540378443 i 5 8 6602540378443 i 1 KUND 3 66 6 5 8 6602540378443 i 5 8 6602540378443 3 4641016151377 amp 10 3 591 50 HUN 5 8 6602540378443 1 1 5 73502631896E 12 4 2264973081038E 13 2 piecewise undefrundeft undef undefzundef undef z undef u det undef lt undef red D N L 20 Addit
57. o whatever you think useful with the article or the package itself My new demanding interests ancient scripts do not allow me to make even a vague promise that will be of some help to you George Douros You can download the files and maybe that somebody can adapt the huge package at least partially for the recent DERIVE versions recommend the website demonstrating George s field of interest ancient scripts http users teilar gr g1951d D N L 20 George Douros Differential Equations with Derive p29 Differential Equations with Derive George Douros Technological Education Institute of Larissa Larissa 41110 Greece Abstract This 1s a presentation of a new Derive package for solving Differential Equa tions ODE MTH The aim is to show how to use it and to explain some of the mathematics behind it so that users can expand it or use it as an educational aid The package currently covers 1 and 2 order Ordinary Differential Equations both linear with variable coefficients and nonlinear 1 Introduction The first version of ODE MTH appeared about a year ago and was written for Derive 2 5 Soft Warehouse found it extremely useful and encouraged the author to publicize and make it available to Derive users In the mean time Derive 3 appeared with new programming functions and ODE MTH grew into an entirely new package Nonlinear equations are now covered almost exhaustively singular solutions are found
58. of the order AUTONOMOUS r v simplifies an expression for dv dy given an autonomous equation y r y v with v repre senting y Example 12 d d v 4l 5 AUTOMOMOUS 1 d d UNI 1 6 DSOLVEI GEN 1 v y y c 2 2 v l d UNC 1 y 2 c Bey Pec 87 SOWE y c v Je 1 v v z je 1 d fey Jc Pey Jc FB DSOLVE1_GEN Ce 1 1 x y k CATANG Ce 1 x k Pey ZC Pey J c 9 DSOLVEI_GEN J e 1 l x y k CATANG Ce 1 x k fey 2 c 10 SOLVECATANC Ce 1 x k y accepting some restrictions for the domain LH COS x k 11 y E 2 Example 21 is given in Fuster s paper Example 16 is the last in my row D N L 20 Comments on the Differential Equations p25 13 AUTOHORMOUS y y l1 Ms j LN y LN v c 14 GE v Ly e 15 SOLUTIONS LN y LH v c v Ly e c c 16 DSOLVEI GEN e 1 x y k x e IN y k Cc 17 C X SOLVE x e LN y k y e E In my opinion Jose Luis tools are a great support even in times of DERIVE 6 Referring to the original DERIVE tools requires a lot of more knowledge which is not so bad and of more manipulating wanted to compare DERIVE with the Tis and tried to solve Examples 9 through 24 using TI NspireCAS and the TI 92 and Voyage 200 Example 9 desclve En xsin da0 y
59. on of the interval 1 lt x 1 is considered in simplifying expressions 1 9 and 11 Simplifying 13 included 2 fails again 1 x Kis CL x o User l x x 1 2 Lu 1 x Simp 1 1 x 2 1 x 3 J 1 x Expd 1 l x 4 x e Real 1 1 User 5 x 1 Simp 1 2 JA x 2 J x 1 DS uu re Simp 3 JG x F x If x gt 2 7 false User true unknown 8 F x true Simp 7 D N L 20 DERIVE USER FORUM p gt G x If x 2 gt 2 9 false User true unknown 10 G x true Simp 9 1 x 11 u IFI CA x 1 x J gt 3 false true unknown User 1 x 12 u true Simp 11 2 1 xX 13 v IF JA x gt 3 false true unknown User l x 14 v unknown Simp 13 Albert Rich 1 Dear Josef Enjoyed reading the DUG Newsletter 19 You have surpassed the 18 Newsletters Soft Warehouse published back in the muMATH days In your response to Wolfgang Propper s question concerning the simplification of radicals you wondered why DERIVE did not simplify his example to 1 x even if x is de clared an element of 1 1 The reason is that DERIVE does not simplify SQRT 1 x 2 to SQRT 1 x SQRT 1 x because this requires rational factoring of 1 x42 DERIVE only tries square free factoring on the argument of radicals because rational factoring can take a very long time e g try rational factoring 1000 x 4 x43 1323 To prop
60. oreover the trivial solution is said to be uniformly asymptotically stable if there ex ists a number A gt 0 such that for all amp lt A the condition lim u t T 0 uniformly for all t holds In this paper we deal with the linear system with constant coefficients in the form x cx 3 where A is a real constant n n matrix In this case we can apply the following assertions Theorem 1 The trivial solution of system 3 is uniformly asymptotically stable iff the real parts of all eigenvalues of matrix A are negative Let P x a ax a x a x is a given polynomial with real coefficients a a 0 0 0 a a a a O The matrix Ayn D nz An Theorem 2 Hurwitz Criterion n21 a gt 0 a 0 4 is said to be the Hurwitz matrix of polynomial 4 a n Real parts of all roots of the polynomial 4 are nega tive iff all main diagonal minors of the Hurwitz matrix are positive It means that D a gt 0 The following sequence is the Hurwitz criterion realised with DERIVE 1 a POSITIVE x If x U 2 1 g MAIM MINOR a k ni Ifk 2n 3 a MINOR MAIM MIMOR a k 1 k 1 k 1 4 Olx a CHARPOLY a x Pix vales If Q D al D 5 Qix a Q x a 6 n DIMENSION a d 1k H Pix a ES dx vec oer VELTOR ITERATE A Kozubik Asymptotically stable Solutions p39 8 zero vec i VECTOR D k n 9 base APPEHD zero vec APPEN
61. printing it in D N L The more contributions you will send the more lively and richer in contents the DERIVE Newsletter will be Preview Contributions for the next issues Graphic Integration Probability Theory Linear Programming Bohm A LOGO in DERIVE Lechner A DREIECK MTH Wadsack AUS IMP Logo and Misguided Missiles Sawada HAWAII 3D Geometry Reichel AUS Parallel and Central Projection Bohm AUS Vector and Vector Indices Sorting Biryukov RUS Algebra at A Level Goldstein UK Tilgung fremd erregter Schwingungen Klingen GER Utility for Complex Dynamic Systems Lechner A Notes on DERIVE 2 6 functions and limits Speck NZL Ski Jump a project with students Scheuermann GER Linear Mappings and Computer Graphics Kummel GER Julia Sets K mmel GER Solving Word Problems with DERIVE B hm AUT and Setif FRA Vermeylen Belgium Lymer FRA Leinbach USA Aue GER Weth GER Wiesenbauer AUT Keunecke GER Weller GER and messages from the derive news mailbase ac uk Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm Herstellung Selbstverlag D N L 20 DERIVE USER FORUM p 3 Glynn D Williams Gwynedd UK Dear Sir have used DERIVE 3 0 for just over a year now and find that it is a significant improve ment over the previous versions both in the functions available and in speed of opera
62. rve d d d 1 x kg er 2 P reCOSCE r SINCt t r SIN d 3 n II I am Lu F J D rt en I 4 men COSC2 t1 2 SINCO SEN 3 3 5 Lummen COSC2 t II Q sINGE SEN 3 3 5 SE 0 P 7 O 0 P With DERIVE 6 we can improve the representation by introducing slider bars for the radius of the circle and for the angle which is formed by the ray PQ and the x axis Thomas Weth A Lexicon of Curves 7 Zur Herleitung der Polardarstellung der Kurve bedarf es zunachst einiger Theorie Gegeben sei eine Kurve x t y t mit dem Kurvenparameter t Gesucht ist zu jedem Kurvenpunkt P der Abstand p zum Ursprung des Koordinatensystems und der Win kel o den die Halbgerade OP mit der x Achse einschlie t vgl Skizze Zu jedem g ben tigt man also zun chst den zugeh rigen Parameterwert t Erh lt man zwischen und eine um kehrbar eindeutige Beziehung mit gewissen Differen zierbarkeitseigenschaften kann man als Funktion von P xit y t o darstellen g und kann die Kurve umparamet 2 risieren Man erh lt dann x 1 o y t oder k rzer 5 x 9 vo Prinzipiell kann man folgenderma en vorgehen o Fur den Winkel gilt 0 1 2 3 4 5 p sean oder tang x t WO x t Diese Gleichung l st man falls es gelingt nach auf und erh lt t in Abh ngigkeit von o also eine Funktion 9 gt t Einsetzen liefert dann x t
63. sin COS 2 2 2 a r sin F r die Kurventangente QT lautet also die Geradengleichung y x tan a Die partielle 2 GE Ableitung nach dem Parameter f hrt zur Gleichung der H llkurve Siehe Erkl rung im Anhang The family of rays with their initial point on a fixed point on a circle form if they reflected at this circle the cardiod Pay attention to the fact that the points of the curve are exactly the osculation points of the tangents Let P the fixed point on the circle the origin of the rays Q is the intersection point of the ray and the circle Then QT is the reflected ray with slope m In triangle OQT we apply the sine rule to obtain the y intercept n of the reflected ray Using DERIVE we find the equation of the family of lines I with angle a as parameter Partial dif ferentiation wrt to the parameter Il leads to the equation of the envelope See appendix and the accompanying file p50 Thomas Weth A Lexicon of Curves 7 D N L 20 Formt man nun mit DERIVE um so liefert die Multiplikation mit cos l I F x yc y eos x sin r sin 0 of II 3y ee NE OO 2 2 2 L st man diese Gleichungen nach x und y auf so erh lt man nach mehreren Versuchen f r die Ein stellungen des Simplification Mode f r Trigonometry und TrigPowers die Koordinaten der Kurven punkte als x Q cosa cos2a und y 2Q sin a sin 2a Damit liefert DERIVE aus der Parameterdarstellung die gesuchte Ku
64. table APPEND table v l viz v43 n 3 n ak 3 2 m_meth 1000 x 0 5 0 0 5 12 Ee pur 9 987121004 10 00002340 10 10 3 10 00002340 10 10 11 10 10 10 12 3 1000 x 4 m meth 0 5 0 0 5 5 x 10 xl xz x3 p 0 5 0 0 5 0 0 0 5 5 6600254037 L 1 5 0 5 5 5 550254037 L S amp 6602540s 2 2 5 amp 6602540s 2 5 amp 600254057 2 5 amp 6602540s 2 3 5 amp 6602540s 2 5 8 65025403 7 c 5 amp 6602540s 2 4 5 amp 66002540s 2 5 amp 60025405 2 5 amp 6602540s7 2 B The next equation has only complex zeros See the first steps 4 d 10 m_methix 7 x 10 10 0 10 20 4 2 11 FIRSTEREWERSEIm_methix Zens 10 10 0 10 20022 104 118 128 12 ES 1 414213562 4 0543395d44 10 1 414213562 2 132092571 10 1 414213562 16 24 4 2 x x 10 13 FIRST REVERSE m_meth _ 10 10 20 x 1 414215562 2 14 b 55355831 10 1 414213561 3 6865475968 10 1 414213561 1 0358845 10 l 414213561 7 Neglect the tiny real part then the solutions are iV2 iN2 iN5 iV5 In the next example the imaginary part can be taken as zero p44 Additional Comments on M ller s Method x d mmethte x 10 0 10 205 205 21 722 18 15 D N L 20 28 O 70346742 1 4581314 10 i 0 70346742 2 7687403 10 i 0 70346742 6 5523512 10 i
65. tion But feel that there are still some loose ends which need tidying up 1 It seems that the configurable menu system was added as an afterthought because no error checking is done on the menu items to see that a the menu tree is workable with all brackets and quotation marks correctly matched up b all the menu items on a particular branch of the menu tree have distinct hotkeys c there is a way out using Quit or its alias Any of the above kinds of error can cause the program to simply hang up requiring a re boot This is unacceptable errors should be indicated with a chance to escape use a modified menu myself because like Substitute and Renumber frequently wanted but buried two levels down the menu tree to be one key operations also like Save Load and Merge to be near the top of the menu 2 There are some bugs in the code factor returns 1 instead of making this function unworkable when used with the factors function Of course one could write one s own factor function e g myfactor a if a factors a to squash the bug but it is messy and such patches should be unnecessary 3 The mouse could be usefully employed to select areas of graphs or expressions to be cop ied pasted or zoomed 4 One area which needs to be attended urgently is the presentation of brackets in multi line expressions These are currently always represented as square this can cause ambiguity if the f
66. together with primitives and a wide class of nonhomogeneous 2 order linear equations with variable coefficients that lead to solutions in terms of Bessel Kummer and Hyper geometric functions is treated in detail The package as well as this article is based on Derive 3 The User Manual DERIVE Version 3 is the main reference Other useful sources are listed in the Additional Resources appendix on pages 331 340 of the Manual 2 The General Purpose Function ODE w x y yl y2 There are over 280 new functions defined in the package Most of these are auxiliary local functions Some can be used independently All are however auxiliaries to a single function which solves ODEs of the 1 and 2 order This function in terms of the default variables is ODE CW X Vy Vis v2 where w is any equation or function involving the independent variable x the unknown dependent variable y y x and its derivatives yl y x and y2 y x For example to solve the equation y x y x 3y x 2x cosx enter ODE y2 y1 3 y 2 x 2 COS x or ODE C qe n ES E p30 George Douros Differential Equations with Derive D N L 20 To override the default variables in ODE all arguments must be entered explicitly For exam ple to solve the equation u t u t 3 enter ODE S TTD 2 35 by Uy L or ODE U FUSCO reru y Many of the methods in the package use solutions of algebraic and or transcendental equa tions It
67. ty OSU und Bernhard Kutz ler von SWHE zwei TI 92 Spezialisten haben angeregt eine TI 92 Kolumne in den DNL aufzu nehmen Sie haben auch versprochen die ersten Beitr ge zu liefern und Anfragen zum TI 92 zu beantworten Das klassische DERIVE f r den PC wird dabei weiterhin nicht zu kurz kommen Die Beitr ge dieser Ausgabe sind schwerpunkt m ig den Differentialgleichungen gewidmet Das ist vielleicht f r manche von Ihnen zu einseitige Kost Ich kann Ihnen Abwechslung ank ndigen 1996 wird es einen ausgesprochen sportlichen DNL geben eine Projektarbeit ber das Schi springen eine Geometrie des Fu balls Baseball ein Vergleich von sportlichen Leistungen und vielleicht ein Beitrag DERIVE und das Tennis netz versprechen allerhand Als kleines Weihnachtsgeschenk finden Sie unter den Dateien im Unterverzeichnis lt STEREO gt einige Stereogramme an denen Sie Ihren magi schen Blick testen k nnen Laden Sie die Bilder in ein Bildbetrachtungsprogramm zB IRFANVIEW Sie lassen sich ausdrucken oder am Bildschirm betrachten Ich w nsche Ihnen viel Spa damit Die Bilder wurden mit POPOUT LITE erstellt Geom Figuren Geckos DUG Geckos SWHE Logo DERIVE in Rahmen und Gecko DERIVE Bitte vergessen Sie nicht Ihre Mitgliedschaft rechtzeitig zu erneuern wenn Sie der DUG die Treue halten wollen Damit bleibt mir nur noch Ihnen und Ihrer Fami lie in meinem und im Namen meiner Frau alles Sch ne f r
68. uation xy 4x 22yy which as given by ODE is x aq y ss ZC a 2 4x 0 enter ELIMINATE x a cl y cl_ a 2 4 2 y 2 4 x 2 0 a e TRANSFORM X w t x x y yl y2 wil transform the equation w when x gt t x To transform for example the equation y 4 cot x y Acsc x y 20 by letting x arccos x enter TRANSFORM X y2 COT x y1 4 CSC x 2 y ACOS x e TRANSFORM Y w Q x y x y yl y2 wil transform the differential equation w when y x O x Y x To transform for example the equation y x 1 y 0 by letting deed ye Y x enter TRANSFORM Yqy2se x59 er L2 e TRANSFORM P w x y yl1 0 r r1 will transform the 1 order differential equation w when x y x gt r 0 cos 0 sin To transform for example the differential equation D yy 2 Gc yy Y enter TRANSFORM P yl 241 x y 2 x y y1 2 e RICCATI w S x x y yl will return a solution of the Riccati DE w which is more general than a known particular solution s x For example if y 1l is a known solution of the Riccati e quation y SS y to find a more general one enter RICOATICVITMIZX 2TyvEsxev 2 l x e GENERAL w lst x 2nd x x y yl y2 wil return the general solution of w p x y q x y r x y f x 0 if one or two solutions of the homogeneous equation are known If two solutions are known they must be entered as a vector 1st x 2nd x If one solution is known it may be entered
69. unction definition contains vectors or if the header accepts them as parameters The Newsletter suffers from this it is sometimes difficult to decide whether a bracket or a pa renthesis should be keyed when typing in code The explanation in the manual is that this is unavoidable because of the limitations of the IBM compatible character set 5 How about a proper ring or spiral bound manual so that one can open the book out flat one has both hands in use when working at a keyboard and a book which tends to close is a nuisance in these circumstances Yours faithfully Glyn D Williams DNL I sent your ideas to Soft Warehouse and I am sure you will receive an answer In my DERIVE versions 3 04 and 3 06 factor returns Concerning point 4 of your complaints I must admit that you are right two souls are fighting in my breast The expressions written in DERIVE syn tax are easy to type in but they often don t represent the mathematical contents as clear as wanted I will pay more attention to this aspect in the future Maybe it will be possible to mix the two forms of representing DERIVE expressions Martin Lindsay Footscray Campus Melbourne Australia am a beginning mathematics education PhD student who is interested in using DERIVE as a project for my research teach upper secondary lower tertiary students 17 21 year olds mainly precalculus and calculus topics would like to hear from anyone who has come p 4 DERIVE
70. ve the idea of trying the same trick once more 4 13 16 2 8 9 15 Z FZ Fz Z EX Er BE ME Fam DE 2 2 16 k 1 RED a b Z k 1 A l l l Therefore a and b are both roots of the quadratic equation x a b x ab x ex SE 0 p58 Johann Wiesenbauer Titbits 6 D N L 20 Again we leave it to DERIVE to calculate its solutions 2 1 SOLUTIONSIx e x 0 x 4 En 8 8 8 8 8 8 ea 25 1173 J17 1 Ste 723 473 T 1 By applying APPROXIMATE one finds out that the first solution is a and the second one is b 1 024740588 0 2439641824 APPROX lim COS p COS 4 9 D 1 02474 p22 n 17 In analogous manner the values of c and d are calculated 3 5 14 12 6 7 SE 10 zZ Z Z Z o Z Z Z C Biss 2 2 16 k 1 RED c d 5 zZz d Er ab 4 Jl al Mea eS a 4 4 di a SOLVE x f x z d A J 34 2 17 J17 1 JG4 2 17 J17 i E I H Ml Y EE SS m 8 8 8 8 8 8 etai SE S Sek 1 HF ke S gir Ji x 1 XI j 32 32 8 8 ec 32 8 8 x 0 172075 x 1 45285 APPROXC lim CCOSC3 g COSCS g 0 172075 q22 n 17 The rest is a piece of cake Because of l 16 l 4 13 l 5 14 20 29 coso cos4 z4 zZ gt z z Iz z z z J 9 cos4p zz z 2 o l z tz z z cos 39 cos5g 4 2 2 we have x cos x cos4g x cosp cos
71. ve with DERIVE in polar form p52 Thomas Weth A Lexicon of Curves 7 D N L 20 Performing this procedure with the given parameter form we obtain a quartic with the vari able cos a which even DERIVE is unable to solve But after performing a translation of the curve such that its vertex is laying in the origin the next attempt turns out to be successful and we can plot the cardiod in polar form see below Der zugeh rige DERIVE Polar Plot is in nebenstehender Abbil dung dargestellt Im Nachhinein ist man immer schlauer Trigonometry Expand Trigpower Auto 2 Z digaigch Z digaigch zr SING z r SINCE COSC y 3 3 3 3 2 r COS Li COSl SINCE 3 Wenn man nun beachtet dass allgemein f r die kartesische und die Polardarstellung immer gilt x 9 y 9 p cos o psin o erh lt man durch Vergleich p Zu cos a Afterwards it looks so easy to obtain the parameter form We should know the general rela tion between the rectangular and the polar form x q voll p cos o psin o and then find by comparison p 1 cosa Da die Kurve sicher eine algebraische ist sie wurde elementar mit Zirkel und Lineal punktweise er zeugt ermitteln wir ihre algebraische Gleichung Der Einfachheit halber verzichten wir auf den Streckfaktor in der Parameterdarstellung und ver wenden x y 2 2r cosa 1 cosa 2r sina 1 cosa Nun ist ein Polynom F x y r mit F x y r 0 zu
72. x inapplicable 2 e DSOLVE1_GEN SEC y TAN x SEC x TAN y inapplicable DSDLUE1 GEM 1 x C3C y x y inapplicable 2 Z DSOLVE1_GEN SIN y COS y COS x inapplicable Before following Llorens Fuster s very valuable suggestions from 1995 let s look at DERIVE s solutions from today Josef The way how DERIVE 6 solves these differential equations 5 DSOLVEl GENY x 4 DSOLVEl GEN v l x2 dv Dix CLNGO 2 v LN x2 x C d DSOLVEl GEN SC Gei TAN x S C oi vfANtGeii CSINGy 2c SIN x E cox 1 DSOLVEl GENL x C5C v X ey yso SINGY Noah x 2 2 y i DSOLVEl GEN SIN y COS y COSCO J TAN x d 2 COS Cy The TI 92 Voyage 200 and TI NSpireCAS are performing pretty the same ee DENN tanly coslx et sinlx N l sinix deSolve Lo SR y x dix x O cx deSolvelx y 1 x xy y 2 y Inly In x x c3 2 f E 2 sec x tanly 4 2 deSolvelx y y 1 x escly x y sinly y cosly Inlx 4c5 X desolvely cos x sin by ICH cosh A Inita n 2 1 2 B c6 coslx sinlx cosy cos x We continue with Llorens Fuster from 1995 J L Fuster Improvements on the Resolution of ODEs The way to solve this problem is suggested from the first example In all these cases DERIVE seems to recognize more easily that the differential equation is separable if it is writ
73. y FALTAXV u v Selec x INT 2f u y k 1 2 vy c y INT 2 u y k 1 2 x c In this way we obtain two equivalent expressions of the general solution It 1s possible that one of the two solutions leads to an unsimplificable integral depending on the form of the equation for these reasons we present the two expressions Finally for the equations of type c and d ie y u x y or y u y v respec tively we have to define in the first step two auxiliary functions with the following purpose to solve the first order differential equation obtained after y v as we stated previously to solve for v in the general solution of this differential equation AUl u v x y k SOLUTIONS DSOLVEl GEN u 1 x v Kk v AU2 u v x y k SOLUTIONS DSOLVE1 GEN u v 1 y v k v I had to adapt the version from DERIVE 3 to DERIVE 6 because of another form of output for the SOLVE command Now we define the corresponding main functions in which we apply the DSOLVEI GEN function in the obtained solution with each auxiliary function As it is pos sible to obtain two or more different expressions when we solve for v we use the VECTOR command FALTAY u v x y K c VECTOR DSOLVEL GEN AUL U V3XV R dir eer OS E DIM AUl u v x y k FALTAX u v x y k c VECTOR DSOLVE1 GEN AUZ u v X y k 1 1 X Y C 1 DIM AU2 u v x y k The efficiency of these functions depends on the following thr
74. y1 y2 are left unchanged only in the global functions ELIMINATE RICCATI GENERAL ODE and the TRANSFORM utilities to reduce typ ing If one uses only the default variables it suffices for example to enter ODEL CY Lay xy 2 rA instead of ODE yl y x 2 x y2 x y Y1 Y2 3 Equations of the 1 Order 3 1 Equations linear in y x The package uses the standard techniques but also looks for symmetries that allow simplify E is invariant under the ing transformations For example the equation f SER D N L 20 George Douros Differential Equations with Derive p31 transformation x y a XOU y This suggests that the transformation y x Y will simplify the original DE If we transform the solved form of the above equation E f 2 we get x X 1 TRANSFORM Y x yl x n f y x n x n y 2 yl xtn y f y df which is a simple separable equation ODE finds the appropriate n by computing n 4 This is all dy 1 We SESCH done automatically without user intervention Take for example the DE y 3 ODE yl 1 y V x y 2 x 2 y Vx y 2 x 4 cl X Extensive search for integrating factors of various forms is performed to make equations exact Take for example the equation x y 4 x 14 y y 20 which is not exact 5 EXACT 1ST x 2 y 3 x lty 2 y1 6 x try 1t and solves the equation l ODE however finds an integrating

THE DERIVE - NEWSLETTER #20 USER GROUP

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