Consider the initial value problem

(solution: y = 1/[2.5 - S(x)] + 0.01x² where S(x) is the Fresnel integral (38).(a) Solve (17) by Euler, improved Euler, and RK methods for 0 ? x ? 5 with step h = 0.2. Compare the errors for x = 1, 3, 5 and comment.

(b) Graph solution curves of the ODE in (17) for various positive and negative initial values.

(c) Do a similar experiment as in (a) for an initial value problem that has a monotone increasing or monotone decreasing solution. Compare the behavior of the error with that in (a). Comment.

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(17) y'= (y 0.01x²)² sin (x²) + 0.02.x, y(0) = 0.4