Question

Math 216 Homework webHW9, Problem 8 Solve the initial value problem " 12x 36x -(t - 5) - 6), x(0) -1,x' (0) 1 (or2 0).

Answer 1

Given IVP is x^11 + 12 x^` + 36 x = delta (t - 5)- delta (t - 6), x(0) = 1, x^1(0) = 1. Rightarrow (1) Apply lplace transform both sides L[x^11 + 12 x^` + 36 x] = L[delta (t - 5)] - L[delta (t - 6)] L(x^11) + 12 L(x^1) + 36 L(x) = e^-5 s - e^-6 s [s^2 x^- s x(0) - x^1 (0)] + 12 [s x^- - x(0)] + 36 x^- = e^-5 s - e^-6 s (s^2 x^- - s - 1) + 12 (s x^- - 1) + 36 x^- = e^-5 s - e^-6 s x^-(s^2 + 12s + 36) - s - 1 - 128 = e^-5 s - e^-6 s x^- = (e^-5 s - e^-6 s + s + 13)/(s^2 + 12s + 36) Apply inverse Laplace transform both sides L^-1 [x^- (s)] = L^-[e^-5 s/(s + 6)^2 - L^-1[e^-6 s/(s + 6)] + L^1 [s + 3/s^2 + 12s + 36] x(t) = L^-1 [e^-5 s/(s + 6)] - L^-[e^-6 s/(s + 6)^2] + L^1 [s + 6 + 7/(s + 6)^2]