Casio FX2.0 Owner's Manual
Contents
1. 2 1 Differential Equations of the First Order 2 Differential Equations of the First Order E Separable Equation Description To solve a separable equation simply input the equation and specify the initial values dyldx fix g y Set Up E i From the Main Menu enter the DIFF EQ Mode Execution 2 o NOOA OQ Press F1 1st to display the menu of first order differential equations and then select 1 Separ Specify f x and g y Specify the initial value for xo yo Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 9 Make V Window settings Press F6 CALC to solve the differential equation PEL 2 2 Differential Equations of the First Order eeeee Example To graph the solutions of the separable equation dy dx y 1 Xo 0 yo 0 1 5 S x 85 7 0 1 Use the following V Window settings Xmin 6 2 Xmax 6 2 Xscale 1 Ymin 3 1 Ymax 3 1 Yscale 1 Procedure CD MEN DIFF EQ F5 SET 2 Output F4 INIT Es F1 1st c1 Separ sur FTN V Window WE OW C J 2 e mw Y LJ 2 C2 CI e 8 C J 2 E 0 Ex o E C wr OC CU e 3 C e OBO 1 9 FS SET 1 Param 3 CJ C 69 OBE G E E9 5 Ex F CALC 92. Result Screen Xscale 1 Yscale 1 FS SET 2 Output FA INIT 6 9 oery V Window 1 OW 0 eO 3 CJ 0 3 C 1 es 1 e sq F CALC Mien Window et iB 09523809 3 1 zcale i IMITITRIGISTDIETOTRCLI TRRaCEIZODIEKTCHG zLUI IF1crI PEL 4 1 Differential Equations of the Nth Order 4 Differential Equations of the Nth Order You can solve differential equations of the first through ninth order The number of initial values required to solve the differential equation depends on its order e Enter dependent variables y y y y y as follows y um G Y o assssereeetreen s F3 ve CD Y 1 PE F3 vm 2 Y2 y oy F3 v 3 Y3 YO ausus F3 vo 8 Y8 LEE F3 Y 9 Y9 E Differential Equation of the Fourth Order The following example shows how to solve a differential equation of the fourth order ye f x Ne y Set Up Bl 1 From the Main Menu enter the DIFF EQ Mode Execution Press F3 N th Press F3 n 4 to select a differential equation of the fourth order Specify y Specify the initial value for xo yo y o y o and yo Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 10 Make V Window settings 11 Press
3. amp 8 Ex 1 E E MC CD E eg E CALC Result Screen 3 1 Linear Differential Equations of the Second Order 3 Linear Differential Equations of the Second Order Description To solve a linear differential equation of the second order simply input the equation and specify the initial values Slope fields are not displayed for a linear differential equation of the second order y f x y gQ y A x Set Up E i From the Main Menu enter the DIFF EQ Mode Execution o N Oa FW Iv Press F2 2nd Specify f x g x and h x Specify the initial value for xo yo y o Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 9 Make V Window settings Press F6 CALC to solve the differential equation PEL 3 2 Linear Differential Equations of the Second Order Example To graph the solution of the linear differential equation of the second order y 9y sin 3x xo 0 yo 1 y o 1 0 S x lt 10 h 0 1 Use the following V Window settings Xmin 1 Xmax 11 Ymin 3 1 Ymax 3 1 Procedure MEN DIFF EQ F2 2nd 0 bx 8 e9 sin 3 K67 Ex x G e C1 e9 F5 SET 3 Param 0 bx 1 0 ES 9 C CD 69 E IParamet er
4. CJ C amp Eg Result Screen Xo yo 0 1 xo yo 0 0 To graph a family of solutions enter a list of initial conditions 2 8 Differential Equations of the First Order E Linear Equation To solve a linear equation simply input the equation and specify initial values dyldx f x y g x Set Up B i From the Main Menu enter the DIFF EQ Mode Execution 2 oo N DA fF o Press F1 1st to display the menu of differential equations of the first order and then select 2 Linear Specify f x and g x Specify the initial value for xo yo Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 9 Make V Window settings Press F6 CALC to solve the differential equation PEL 2 4 Differential Equations of the First Order eeeee Example To graph the solution of the linear equation dy dx xy x Xo 0 yo 2 5 S x 5 h 0 1 Use the following V Window settings Xmin 2 6 2 Xmax 6 2 Xscale 1 Ymin 2 3 1 Ymax 3 1 Yscale 1 Procedure E DIFF EQ F5 SET 2 Output FA INIT E F1 1st 2 Linear sary FTN V Window Kan Ex c3 6 C 2 Eg ker Ex 6 C 2 DEY DITO c3 2 e OBOE FS SET Q Param 3 C 1 es OBE
5. Press Ex Yes The entered differential equation is converted to a system of three first order differential equations Initial values are also converted accordingly PEL 4 4 Differential Equations of the Nth Order eeeee Example Express the differential equation below as a set of first order differential equations y sinx y y xo 0 yo 0 y o 1 y o 0 Procedure Cp ww DIFF EQ 2 F3 N th amp F3 n 3 eS sin FA O F3 y O O F3 ye 2 e Me 0 e 1 e 0 e F2 2SYS PS Yes The differential equation is converted to a set of first order differential equations as shown below y1 dyldx y2 y2 dy dx ys ys sin x yz y3 Initial values are also converted to xo 0 y1 o 0 yz o 1 and ys o 0 Result Screen 1st Order S stem 3 305 7n DELA SET ear 3t On the system of first order differential y gt Y1 equations screen dependent valuables are y gt Y2 expressed as follows ys gt Y3 5 1 System of First Order Differential Equations 5 System of First Order Differential Equations A system of first order differential equations for example has dependent variables y1 yz eds and ys and independent variable x The example below shows a system of first order differential equations y1 ye y2 2 y1 sin x Set Up B 1 From the Main Menu enter the DIFF
6. differential equation you need to make V Window settings Xmin x axis minimum value View Window max x axis maximum value scale x axis value spacing dot value corresponding to one x axis dot Ymin y axis minimum value max y axis maximum value scale y axis value spacing 8 Press F6 CALC to solve the differential equation The calculated result is graphed or stored in the list 3t Only the slope fields are displayed if you do An error occurs if you input variable y in the not input initial values or if you input the wrong function f x Variable x is treated as a type of initial values variable Other variables A through Z r 0 An error occurs if you set SF to zero and you excluding X and Y are treated as constants do not input the initial values or if you input and the value currently assigned to that the initial values inappropriately variable is applied during the calculation 3t You are advised to input parentheses and a An error occurs if you input variable x in the multiplication sign between a value and an function g y Variable y is treated as a expression in order to prevent calculation variable Other variables A through Z r 0 errors excluding X and Y are treated as constants 3 Do not confuse the key and the key and the value currently assigned to that A syntax error occurs if you use the key variable is applied during the calculation as the subtraction symbol PEL wee
7. F6 CALC to solve the differential equation o d DOA FW Inm PEL wee 4 2 Differential Equations of the Nth Order eeeee Example To graph the solution of the differential equation of the fourth order below y 0 xo 0 yo 0 y o 2 y a 0 y 3 5b S x 85 h 0 1 Use the following V Window settings Xmin 6 2 Xmax 6 2 Xscale 1 Ymin 3 1 Ymax 3 1 Yscale 1 Procedure Cp M DIFF EQ 9 F5 SET 2 Output F4 INIT Esc F3 N th oer V Window amp F3 n 4 by OOO 0 P 6 O 2 e 0 P 1 eO 0 e 3 CJ 00 2 BOW 9 0 Ex 1 Ge Ec 3 e FS CALC F5 SET 3 Param OGE 5 e9 6 1 e Eg IPar ameter iew Window le i ie laa IHITITRIGMsTOISTOIRGL Result Screen PEL E Converting a High order Differential Equation to a System of the First 4 3 Differential Equations of the Nth Order Order Differential Equations You can convert a single N th order differential equation to a system of n first order differential equations Set Up E i From the Main Menu enter the DIFF EQ Mode Execution N 3 2 3 4 Press F3 N th Press F3 n 3 to select a differential equation of the third order Perform substitutions as follows y gt Y1 E3 v0 CD y 2 Y2 F3 v9 2 y gt Y3 F3 Om 3 Specify the initial value for xo yo y o and y o Press F2 CSYS
8. Order Differential Equations ee oc o Example2 To graph the solution of the system of first order differential equations below 1 2 03 6 2 2 y1 3 y2 Xo 0 y1 o 1 y2 o 1 4 OS x 10 h 0 1 Use the following V Window settings Xmin 1 Xmax 11 Xscale 1 Ymin 1 Ymax 8 Yscale 1 Procedure Cp MEN DIFF EQ 9 F5 SET 2 Output F4 INIT F4 SYS FI SEL F3 2 Select y and y2 to graph amp OB C E30 2 C 9 3 7 EZ LIST L1 Be Select LIST om to store the values for x in LIST1 F3 LIST E Select LIST2 to CX 2 39 F3 yn 0 CJ 3 store the values for y in LIST2 3 F3 yn 2 e F2 LIST 3 E5 Select LISTS to 9 3 store the values for y2 in LIST3 o e ESC CU G9 4 Eg EN V Window ESET Param oomommnme Wes CD 6 C ES ow FS CALC 6 1 e Eg Parameter ist ana ard uth 5th SEL ILISTIrarmI IIT EcL Result Screen PEL ey 5 4 System of First Order Differential Equations E Further Analysis To further analyze the result we can graph the relation between y1 and y2 Procedure WEN STAT List 1 List 2 and List 3 contain values for x y and yz respectively FI GRPH 5 Set FJ GPH1 F2 xy g FALIST 2 ES XLIST LIST2 y1 GRPHIGALCITES A amp 1 D FA LIST 3 E YLIST LISTS y2 E Fi GRPH 1 S G
9. 1 E E 8 P F6 CALC 6 C Q0 6 amp 9 Eg Result Screen 2 5 Differential Equations of the First Order E Bernoulli equation To solve a Bernoulli equation simply input the equation and specify the power of y and the initial values dyldx f x y g 9 y Set Up E i From the Main Menu enter the DIFF EQ Mode Execution 2 o NOOA OQ Press F1 1st to display the menu of differential equations of the first order and then select 3 Bern Specify f x g x and n Specify the initial value for xo yo Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 9 Make V Window settings Press F6 CALC to solve the differential equation PEL 2 6 Differential Equations of the First Order eeeee Example To graph the solution of the Bernoulli equation dy dx 2y y xo 0 yo 1 5 S x S 5 h 0 1 Use the following V Window settings Xmin 2 6 2 Xmax 6 2 Xscale 1 Ymin 2 3 1 Ymax 3 1 Yscale 1 Procedure M DIFF EQ OO es amp g F1 1st 3 Bern F5 SET 2 Output F4 INIT Esc Goja EA GH V Window 5 D 69 9 8 CJ 2 E93 2 P 6 C 2 Ex Me D eO 1 e 3 CJ 00 FB SET 3 Param 3 CJ C E9 OB 1 Ex Esc 8 E Resu
10. Differential Equations DIFF EQ Software for the ALGEBRA FX 2 0 Using the DIFF EQ Mode Differential Equations of the First Order Linear Differential Equations of the Second Order Differential Equations of the Nth Order System of First Order Differential Equations a Aa 0 Nx 1 1 Using the DIFF EQ Mode 1 Using the DIFF EQ Mode You can solve differential equations numerically and graph the solutions The general procedure for solving a differential equation is described below Set Up 1 From the Main Menu enter the DIFF EQ Mode Differential Eauation Fi ist Order Eauation F2tend Order Lin Ea E3 Hth Order Eauation F4i ist Order System F2 Recall Last ist and TH tnTe vs TREL Execution 2 Select the differential equation type F1 1st Four types of first order differential equations F2 2nd Second order linear differential equations F3 N th Differential equations of the first order through ninth order FA SYS System of the first order differential equations F5 RCL Displays a screen for recalling a previous differential equation e With F1 1st you need to make further selections of differential equation type See Differential equations of the first order for more information e With F3 N th you also need to specify the order of the differential equation from 1 to 9 e With F4 SYS you also need to specify the number of unknowns from 1 t
11. EQ Mode Execution 2 Press F4 SYS 3 Enter the number of unknowns 4 Enter the expression as shown below y gt Y1 E369 2 O gt Y2 Mom Qe gt Y9 o0 Specify the initial value for xo y o y2 o and so on if necessary Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 10 Make V Window settings 11 Press F6 CALC to solve the system of first order equations for y y2 and so on Oo O O PEL 5 2 System of First Order Differential Equations eeeee Example 1 To graph the solution of first order differential equations with two unknowns below 1 2 y2 2 y1 sin x xo 0 y1 o 1 y2 o 0 1 2 Sx 5 h 0 1 Use the following V Window settings Xmin 3 Xmax 6 Xscale 1 Ymin 2 Ymax 2 Yscale 1 Procedure iN DIFF EQ FB SET 2 Output FA INIT F3 SYS FI SEL F2 2 Select y and yz to graph F3 yr 2 e ESC E3 yn 1 C Gin oer V Window 0 9 OB P9 10 Ex 6 e5 6 C J 1 e Q0 P9 FS SET G Param c3 2 p OBE 2 Ex 8 Ex O amp Eg 6 G e Eg FS CALC Ist 2r d Jrd uth 5th Parameter SEL ILISTIrarmI IIT REL Result Screen PEL wee 5 3 System of First
12. erential equation calculation is performed Before the calculation DIFF EQ stores the latest graph functions into the currently specified Graph Memory G Mem location After the calculation it recalls the graph functions from the specified G Mem location without deleting the G Mem data Because of this you should specify the G Mem location number where the DIFF EQ Mode stores the graph functions PEL
13. lt Screen CALC 2 7 Differential Equations of the First Order Bl Others To solve a general differential equation of the first order simply input the equation and specify the initial values Use the same procedures as those described above for typical differential equations of the first order dyldx f x y Set Up E i From the Main Menu enter the DIFF EQ Mode Execution 2 o N Oo A OQ Press F1 1st to display the menu of differential equations of the first order and then select 4 Others Specify f x y Specify the initial value for xo yo Press F5 SET 1 Param Specify the calculation range Specify the step size for h Press F5 SET 2 Output Select the variable you want to graph and then select a list for storage of the calculation results 9 Make V Window settings Press F6 CALC to solve the differential equation PEL 2 8 Differential Equations of the First Order eeeee Example To graph the solution of the first order differential equation dyldx cos x xo 0 yo 1 5 S x 85 h 0 1 Use the following V Window settings Xmin 2 6 2 Xmax 6 2 Xscale 1 Ymin 2 3 1 Ymax 3 1 Yscale 1 Procedure E DIFF EQ F5 SET 2 Output FA INIT E F1 1st 4 Others sary FTN V Window 3 8 CJ 2 amp 9 DEY O 2 e C1 e 1 eO FS SET 1 Param c3 3 C Q1 3 OBE 3 C 00
14. o 9 3 Enter the differential equation 4 Specify the initial values 5 Press F5 SET and select 1 Param to display the Parameter screen Specify the calculation range Make the parameter settings you want Step size for the classical Runge Kutta method fourth order Number of steps for graphing and storing data in LIST The number of slope field columns displayed on the screen 0 100 The slope fields can be displayed only for differential equations of the first order When graphed for the first time a function is graphed according to a value of Step For always graphed with every step When the example when Step is set to 2 the function is function is graphed again however it is graphed with every two steps PEL 1 2 Using the DIFF EQ Mode 6 Specify variables to graph or to store in LIST Press F5 SET and select 2 Output to display the list setting screen x y y YO y stand for the independent variable the dependent variable the first order derivative the second order derivative and the eighth order derivative respectively 1st 2nd 3rd 9th stand for the initial values in order To specify a variable to graph select it using the cursor keys and press F1 SEL To specify a variable to store in LIST select it using the cursor keys and press F2 LIST 7 Press oer V Window to display the V Window setting screen Before you solve a
15. ph1 Result Screen 1 TRacEIZooMIEKTCHCRLCIDerGI P PEL mee 5 5 System of First Order Differential Equations Important This calculator may abort calculation part way through when an overflow occurs part way through the calculation when calculated solutions cause the solution curve to extend into a discontinuous region when a calculated value is clearly false etc The following steps are recommended when the calculator aborts a calculation as described above 1 If you are able to determine beforehand the point where the solution curve overflows stop the calculation before the point is reached 2 If you are able to determine beforehand the point where the solution curve extends into a discontinuous region stop the calculation before the point is reached 3 In other cases reduce the size of the calculation range and the value of h step size and try again 4 When you need to perform a calculation using a very wide calculation range store intermediate results in a list and perform a new calculation starting from step 3 using the stored results as initial values You can repeat this step multiple times if necessary E SET UP Items G Mem G Mem 20 1 20 Specifies a memory location G Mem No for storage of the latest graph functions Note the following regarding SET UP screen settings whenever using the DIFF EQ Mode The DIFF EQ Mode temporarily stores data into Graph Memory whenever a diff
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