Euler's Method:

ClrHome
ClrAllLists
Input "DY/DX=",Str0
Input "X0=",X
Input "Y0=",Y
Input "STEP SIZE=",H
Input "FINAL X=",Z
(Z-X)/H -> N
For(I,1,N)
expr(Str0) -> F
Y+H*F -> Y
X+H -> X
Output(7,1," ") Put several spaces between the quotes to erase old values.
Output(8,1," ") Put several spaces between the quotes to erase old values.
Output(7,1,X)
Output(8,1,Y)
X -> L1(I):Y -> L2(I)
Pause
End

The answers are stored in lists 1 and 2. To see a plot of the solution:
2nd STATPLOT PlotsOn
Set the proper WINDOW
GRAPH.

Euler slope field plotter:

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ClrDraw
Input "DY/DX=",Str0
Input "Xmin=",Xmin
Input "Xmax=",Xmax
Input "Ymin=",Ymin
Input "Ymax=",Ymax
(Xmax-Xmin)/10 -> W
(Ymax-Ymin)/10 -> V
DispGraph
For(M,Xmin,Xmax,W)
For(N,Ymin,Ymax,V)
M -> X
N -> Y
expr(Str0) -> Z
M-W -> P:N-W*Z -> Q:M+W -> R:N+W*Z -> S
2*((P-R)^2+(Q-S)^2))^ .5 -> K
Line(M-W/K,N-W*Z/K,M+W/K,N+W*Z/K)
End
End
Pause
ClrHome

If you get a divide by 0 error with a fraction, try changing your range values to odd decimals, e.g. -3.07

Runge Kutta method: variables Q,R,S,T are k1,k2,k3,k4

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ClrAllLists
Input "DY/DX=",Str0
Input "X0=",A
Input "Y0=",B
Input "STEP SIZE=",H
Input "FINAL X=",Z
(Z-A)/H -> N
A -> X:B -> Y
For(I,1,N)
X -> V:Y -> W
expr(Str0) -> Q
V+H/2 -> X
W+H*Q/2 -> Y
expr(Str0) -> R
W+H*R/2 -> Y
expr(Str0) -> S
V+H -> X
W+H*S -> Y
expr(Str0) -> T
(Q+2*R+2*S+T)/6 -> K
W+H*K -> Y
Output(7,1," ") Put several spaces between the quotes to erase old values.
Output(8,1," ") Put several spaces between the quotes to erase old values.
Output(7,1,X)
Output(8,1,Y)
X -> L1(I):Y -> L2(I)
Pause
End
ClrHome

The answers are stored in lists 1 and 2. To see a plot of the solution:
2nd STATPLOT PlotsOn
Set the proper WINDOW
GRAPH.

Runge Kutta method for 2nd order differential equations:

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ClrAllLists
Input "DX/DT=",Str0
Input "DY/DT=",Str1
Input "T0=",T Input "X0=",X
Input "Y0=",Y
Input "STEP SIZE=",H
Input "FINAL T=",Z
(Z-T)/H -> N
For(I,1,N+1)
X -> L1(I)
Y -> L2(I)
T -> L3(I)
X -> A
Y -> B
T -> C
expr(Str0) -> D
expr(Str1) -> E
C+H/2 -> T
A+H*D/2 -> X
B+H*E/2 -> Y
expr(Str0) -> L
expr(Str1) -> M
A+H*L/2 -> X
B+H*M/2 -> Y
expr(Str0) -> P
expr(Str1) -> Q
C+H -> T
A+H*P -> X
B+H*Q -> Y
expr(Str0) -> R
expr(Str1) -> S
(D+2*L+2*P+R)/6 -> F
(E+2*M+2*Q+S)/6 -> G
A+H*F -> X
B+H*G -> Y
End

The answers are stored in lists 1 and 2. To see the solutions, STAT | EDIT, L1 is X, L2 is Y, L3 is T.
To see a plot of the solution:
2nd STATPLOT PlotsOn
Set the proper WINDOW | GRAPH.

Phase Plots:

Use Program eulslope. This program will do autonomous (non-time dependent) plots only.
Since dy/dt over dx/dt = dy/dx enter y' as a numerator and x' as a denominator.
For example: If y' = 1-x^2 and x' = x-y enter dy/dx= (1-x^2)/(x-y)
If you get a divide by 0 error, try changing your range values to odd decimals, e.g. -3.07


Graphing the unit step function:
TI83 use the logical operations. Example: U(t-1) + 2U(t-3) - 3U(t-4) is: Y1 = (x>1) + 2(x>3) - 3(x>4)

TI89 Many Differential Equation utilities are built in to the TI89.

Slope Field Plot:

Set MODE Graph DIFF EQUATIONS Y=
y1' = must be a function of y1 and t; For example if dy/dx = x^2 + y^2 with y(0)=2: enter y1' = t^2 + y1^2 with t=0 and yi1=2
then GRAPH
If no field comes the fields may be turned off. Hit the green diamond key and the Y= key; then the green diamond key and the | key; this will give you Graph Formats; go to Fields -> SLPFLD.

Euler's Method:

Set MODE Graph DIFF EQUATIONS
While in Y= do this: Green diamond key, then | key: Solution Method -> EULER; Fields -> FLDOFF
In Y= Define y1' = f( y1,t); in WINDOW enter your starting value and stepsize; in TABLE see the results in columns.

Example: dy/dx = x + y with y(0)=1: from x=0 to 2 by 0.1

Y=: y1' = t + y1; t0=0; yi1=1
WINDOW: t0=0; tmax=1; tstep=0.1
TABLE: gives values in columns
You should get y(.5)=1.721 and y(1)=3.1875
Note: TblSet allows you to start at any t value so you don't have to arrow a long way down the list. Use deltatbl to look at intermediate values so the list doesn't get too long. (This doesn't affect tstep.)
GRAPH to graph these points:

Runge Kutta method:

Set MODE Graph DIFF EQUATIONS
While in Y= do this: Green diamond key, then | key: Solution Method -> RK; Fields -> FLDOFF
In Y= Define y1' = f( y1,t); in WINDOW enter your starting value and stepsize; in TABLE see the results in columns.

Example: dy/dx =-3x^2 y with y(0)=3: from x=0 to 1by 0.25

In the calculator:
Y=: y1' = -3t^2 y1; t0=0; yi1=3
WINDOW: t0=0; tmax=1; tstep=0.25
TABLE: gives values in columns
You should get y(.5)=2.6474 and y(1)=1.1039
Note: TblSet allows you to start at any t value so you don't have to arrow a long way down the list. Use deltatbl to look at intermediate values so the list doesn't get too long. (This doesn't affect tstep.)
GRAPH to graph these points:

Runge Kutta method for 2nd and higher order differential equations:

You must change the 2nd order DE to two first order equations and enter both of them.
Example: y" + 9y = sin(2x); y(0)=1; y'(0) = -1 from t = 0 to t = 3 stepsize h = 0.2
Using standard techniques convert the 2nd order DE to two 1st order DE in y1 and y2
y1' = y2 and y2' = -9y1 + sin(2x)

In the calculator:
Y=: y1' = y2; y2' = -9y1 + sin(2t); t0 = 0; yi1 = 1; yi2 = 0
WINDOW: t0 = 0; tmax = 3; tstep = 0.2
TABLE: gives values of t,y1,y2 in columns which are really x,y,y'
You should get y(0.2) = 0.8278 and y(1) = -0.8247
GRAPH to graph these points which are the solution curve y(x) and y'(x):

Phase Plots:

Set MODE | Graph | DIFF EQUATIONS
Y=; Green diamond key, then | key: Solution Method -> RK; Fields -> DIRFLD
Most books work in x,y, and t. The Calculator works in y1,y2 and t; Rewrite x as y1 and y as y2.
Example: x' = 14x - 2x^2 - xy; y' = 16y - 2y^2 - xy
Y= key; y1' = 14y1 - 2y1^2 - y1*y2; y2' = 16y2 - 2y2^2 - y1*y2
Note: you must use the multiplication key between variables or the calculator thinks it is all one variable. For a typical curve set t0, yi1, and yi2 = to initial values. For example, if x(0) = 1 and y(0) = 3, then t0 = 0, yi1 = 1, and yi2 = 3.

Graphing the unit step function on TI89 is U(t-a) = 1/2(1+sign(x-a)) -> sign() is in the catalog.
TI83 use the logical operations. Example: U(t-1) + 2U(t-3) - 3U(t-4) is: Y1 = (x>1) + 2(x>3) - 3(x>4

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