#04. Consider the integral equation, Ø(t) + f(t − 1) Ø(1) dt = sin2t (1) (a) Solve the integral equation using Laplace t

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04 Consider The Integral Equation O T F T 1 O 1 Dt Sin2t 1 A Solve The Integral Equation Using Laplace T 1 (35.11 KiB) Viewed 30 times

#04. Consider the integral equation, Ø(t) + f(t − 1) Ø(1) dt = sin2t (1) (a) Solve the integral equation using Laplace transforms [5marks] (b) By differentiating equation (1) twice with respect to t, show that Ø(t) satisfies the differential equation Ø"(t) + Ø(t) = -4sin2t. Show also that Øsatisfies the initial conditions Ø(0) = 0; Ø'(0) = 2 [6marks] (c) Solve the initial value problem in part (b) and verify that the solution is the same as the one you obtain in part (a). [9marks] Hint: For pat (b), you need to appropriately make use of the Fundamental Theorem of Calculus after separately writing the two steps within the integral sign.