There are many different ways to write a computer or calculator program that will perform the calculations needed to estimate the values of a solution to a differential equation with an initial or boundary condition. Here are two examples in BASIC that illustrate the simplicity of such programs.
Example IV.Y.1.
Suppose y' = x 2 with y(0) = 3.
Estimate the values of y using Euler's method for x i =
i / 10 where i =1 to 10.
Solution: The following program will perform the desired estimation and display the results in a table showing the values of x i and the estimates for f(x i) as well as the corresponding values of f '(x i) and f '(x i) . dx .
10 LET X = 0
20 LET Y = 3
30 LET DX = .1
40 DEF FNP(X)=X^2
50 FOR N = 1 TO 11
60 PRINT X, Y, FNP(X), FNP(X)*DX
70 LET Y = Y + FNP(X)*DX
80 LET X = X + DX
100 NEXT N
1000 END
Example IV.Y.2. Suppose y' = x - y with y(0) = 3. Estimate the values of y using Euler's method for x i = i / 10 where i = 1 to 10.
Solution: The following program which is quite similar to the previous example will perform the desired estimation and display the results in a table showing not only the values of x i and the estimates for f(x i) but also the corresponding values of f '(x i, y i) and f '(x i, y i) . dx . Only lines 40, 60, and 70 have been changed to allow the derivative to depend on both X and Y.
10 LET X = 0
20 LET Y = 3
30 LET DX = .1
40 DEF FNP(X,Y) = X -Y
50 FOR N = 1 TO 11
60 PRINT X , Y , FNP(X,Y), FNP(X,Y)*DX
70 LET Y = Y + FNP(X,Y)*DX
80 LET X = X + DX
100 NEXT N
1000 END