Casio Differential Equation Quick start guide

Differential Equations (DIFF EQ)
Software for the ALGEBRA FX 2.0
1. Using the DIFF EQ Mode
2. Differential Equations of the First Order
3. Linear Differential Equations of the Second Order
4. Differential Equations of the Nth Order
5. System of First Order Differential Equations

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.
Execution
2. Select the differential equation type.
•1(1st) ........ Four types of first order differential equations
•2(2nd) ...... Second order linear differential equations
•3(N-th) ...... Differential equations of the first order through ninth order
•4(SYS) ..... System of the first order differential equations
•5(RCL) ..... Displays a screen for recalling a previous differential equation.
• With 1(1st), you need to make further selections of differential equation type. See
“Differential equations of the first order” for more information.
• With 3(N-th), you also need to specify the order of the differential equation, from 1
to 9.
• With 4(SYS), you also need to specify the number of unknowns, from 1 to 9.
3. Enter the differential equation.
4. Specify the initial values.
5. Press 5(SET) and select b(Param) to display the Parameter screen. Specify the
calculation range. Make the parameter settings you want.
•h ................... Step size for the classical Runge-Kutta method (fourth order)
• Step ............. Number of steps for graphing*1and storing data in LIST.
• SF ................ 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.
*1When graphed for the first time, a function is
always graphed with every step. When the
function is graphed again, however, it is
graphed according to a value of Step. For
example, when Step is set to 2, the function is
graphed with every two steps.

1-2
Using the DIFF EQ Mode
6. Specify variables to graph or to store in LIST.
Press 5(SET) and select c(Output) to display the list setting screen.
x, y, y(1), y(2), ....., y(8) 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 (f, c) and press
1(SEL).
To specify a variable to store in LIST, select it using the cursor keys (f, c) and
press 2(LIST).
7. Press !K(V-Window) to display the V-Window setting screen. Before you solve a
differential equation, you need to make V-Window settings.
Xmin …x-axis minimum value
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 6(CALC) to solve the differential equation.
•The calculated result is graphed or stored in the list.
# Only the slope fields are displayed if you do
not input initial values or if you input the wrong
type of initial values.
# An error occurs if you set SF to zero and you
do not input the initial values, or if you input
the initial values inappropriately.
# You are advised to input parentheses and a
multiplication sign between a value and an
expression in order to prevent calculation
errors.
# Do not confuse the -key and the -key.
A syntax error occurs if you use the -key
as the subtraction symbol.
# An error occurs if you input variable yin the
function f(x). Variable xis treated as a
variable. Other variables (A through Ζ, r,
θ ,
excluding X and Y) are treated as constants
and the value currently assigned to that
variable is applied during the calculation.
# An error occurs if you input variable xin the
function g(y). Variable yis treated as a
variable. Other variables (A through Ζ, r,
θ ,
excluding X and Y) are treated as constants
and the value currently assigned to that
variable is applied during the calculation.

2-1
Differential Equations of the First Order
2. Differential Equations of the First Order
kSeparable Equation
Description
To solve a separable equation, simply input the equation and specify the initial values.
dy/dx = f(x)g(y)
Set Up
1. From the Main Menu, enter the DIFF EQ Mode.
Execution
2. Press 1(1st) to display the menu of first order differential equations, and then select
b(Separ).
3. Specify f(x) and g(y).
4. Specify the initial value for x0, y0.
5. Press 5(SET)b(Param).
6. Specify the calculation range.
7. Specify the step size for h.
8. Press 5(SET)c(Output).
Select the variable you want to graph, and then select a list for storage of the
calculation results.
9. Make V-Window settings.
10. Press 6(CALC) to solve the differential equation.

2-2
Differential Equations of the First Order
○○○○○
Example To graph the solutions of the separable equation dy/dx = y2–1,
x0= 0, y0= {0, 1}, –5 <<
<<
<x<<
<<
<5, 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
1mDIFF EQ
21(1st)b(Separ)
3bw
a-(Y)Mc-bw
4aw
!*( { )a,b!/( } )w
55(SET)b(Param)
6-fw
fw
7 a.bwi
85(SET)c(Output)4(INIT)i
9!K(V-Window)
-g.cw
g.cw
bwc
-d.bw
d.bw
bwi
06(CALC)
Result Screen
# To graph a family of solutions, enter a list of
initial conditions.
(x0, y0) = (0,0)
(x0, y0) = (0,1)

2-3
Differential Equations of the First Order
kLinear Equation
To solve a linear equation, simply input the equation and specify initial values.
dy/dx + f(x)y = g(x)
Set Up
1. From the Main Menu, enter the DIFF EQ Mode.
Execution
2. Press 1(1st) to display the menu of differential equations of the first order, and then
select c(Linear).
3. Specify f(x) and g(x).
4. Specify the initial value for x0, y0.
5. Press 5(SET)b(Param).
6. Specify the calculation range.
7. Specify the step size for h.
8. Press 5(SET)c(Output).
Select the variable you want to graph, and then select a list for storage of the
calculation results.
9. Make V-Window settings.
10. Press 6(CALC) to solve the differential equation.

2-4
Differential Equations of the First Order
○○○○○
Example To graph the solution of the linear equation dy/dx + xy = x,
x0= 0, y0= –2, –5 <<
<<
<x<<
<<
<5, 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
1mDIFF EQ
21(1st)c(Linear)
3vw
vw
4aw
-cw
55(SET)b(Param)
6-fw
fw
7 a.bwi
85(SET)c(Output)4(INIT)i
9!K(V-Window)
-g.cw
g.cw
bwc
-d.bw
d.bw
bwi
06(CALC)
Result Screen

2-5
Differential Equations of the First Order
kBernoulli equation
To solve a Bernoulli equation, simply input the equation and specify the power of yand the
initial values.
dy/dx + f(x)y= g(x)y
n
Set Up
1. From the Main Menu, enter the DIFF EQ Mode.
Execution
2. Press 1(1st) to display the menu of differential equations of the first order, and then
select d(Bern).
3. Specify f(x), g(x), and n.
4. Specify the initial value for x0, y0.
5. Press 5(SET)b(Param).
6. Specify the calculation range.
7. Specify the step size for h.
8. Press 5(SET)c(Output).
Select the variable you want to graph, and then select a list for storage of the
calculation results.
9. Make V-Window settings.
10. Press 6(CALC) to solve the differential equation.

2-6
Differential Equations of the First Order
○○○○○
Example To graph the solution of the Bernoulli equation dy/dx –2y= –y2,
x0= 0, y0= 1, –5 <<
<<
<x<<
<<
<5, 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
1mDIFF EQ
21(1st)d(Bern)
3-cw
-bw
cw
4aw
bw
55(SET)b(Param)
6-fw
fw
7 a.bwi
85(SET)c(Output)4(INIT)i
9!K(V-Window)
-g.cw
g.cw
bwc
-d.bw
d.bw
bwi
06(CALC)
Result Screen

2-7
Differential Equations of the First Order
kOthers
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.
dy/dx = f(x, y)
Set Up
1. From the Main Menu, enter the DIFF EQ Mode.
Execution
2. Press 1(1st) to display the menu of differential equations of the first order, and then
select e(Others).
3. Specify f(x, y).
4. Specify the initial value for x0, y0.
5. Press 5(SET)b(Param).
6. Specify the calculation range.
7. Specify the step size for h.
8. Press 5(SET)c(Output).
Select the variable you want to graph, and then select a list for storage of the
calculation results.
9. Make V-Window settings.
10. Press 6(CALC) to solve the differential equation.