Runge-Kutta Method The fourth-order approximation invented by Runge and Kutta can be surpr...

Runge-Kutta Method The fourth-order approximation invented by Runge and Kutta can be surprisingly accurate, even with a ridiculously large step size. To see this, for Problem, use the given step size with the IVP

to do the following:

(a) Compute for a single step the Euler approximation, the second-order Runge-Kutta approximation, and the fourth-order Runge-Kutta approximation.

(b) Add the three approximations in part (a)to the graph of the actual solution, as given in Fig. 8, and describe what you see.

Figure 8 Actual solution of y' = t + y, y(ɸ), on [−1, 1]

(c) Verify that y(t) = et − t − 1 satisfies the DE, then calculate the numerical values for the actual solution at y(1) or y(−1).

h = −1.0