Consider the Volterra integral equation (see Problem 23)

Consider the Volterra integral equation (see Problem 23) ?(t) + ? t 0 (t ? ?)?(?) d? = sin 2t. (i) (a) Solve the integral equation (i) by using the Laplace transform. (b) By differentiating Eq. (i) twice, show that ?(t) satisfies the differential equation ???(t) + ?(t) = ?4 sin 2t. Show also that the initial conditions are ?(0) = 0, ?? (0) = 2. (c) Solve the initial value problem in part (b) and verify that the solution is the same as the one in part (a). In each of Problems 25 through 27: (a) Solve the given Volterra integral equation by using the Laplace transform. (b) Convert the integral equation into an initial value problem, as in Problem 24(b). (c) Solve the initial value problem in part (b) and verify that the solution is the same as the one in part (a).