Integral Equation(a) Show that the IVP y' = f(t, y), y(t0) = y0, is equivalent to the...

Integral Equation

(a) Show that the IVP y' = f(t, y), y(t0) = y0, is equivalent to the integral equation

by verifying the following two statements: (i) Every solution y(t) of the IVP satisfies the integral equation; (ii) Any function y(t) satisfying the integral equation satisfies the IVP.

(b) Convert the IVP y' = f(t), y(0) = y0, into an equivalent integral equation as in part (a). Show that calculating the Euler-approximate value of the solution to this IVP at t = T is the same as approximating the right-hand side of the integral equation by a Riemann sum (from calculus) using left endpoints.

(c) Explain why the calculation of part (b) depends on having the right-hand side of the differential equation independent of y.