Question

Differential Equations

Work these problems out on this sheet. I. Compute L(2t4 +e-6t + 18) II. Compute L1 65 8 s2 +6s+8

Answer 1

L(2 t^4 + e^- 6t + 18) = L[2 t^4] + L[e^- 6t] + L[18] = 2L [t^4] + L[e^- 6t] + 18 L[1] = 2.angle 4/s^(4 + 1) + 6.1/(s + 6) + 18.1/s since L [t^n] = angle n/s^n + 1: L[e^at] = 1/(s - a) & L[1] = 1/s] = 48/s^2 + 6/(s + 6) + 18/s L^-1[s/(s^2 + 6s + 8)] partial fraction of s/(s^2 + 6s + 8) = s/(s + 2)(s + 4) = 2/(s + 4) - 1/(s + 2) L^-1[s/(s^2 + 6s + 8)] = L^-1[2/(s + 4) - 1/(s + 2)] = L^-1[2/s + 4] - L^-1[1/s+ 2] \r\n = 2 L^-1[1/s+ 4] - L^-1[1/s+ 2] = 2 e^- 4t - e^- 2t [since L^-1[1/(s - a)] = e^at L^-1[s/(s^2 + 6 s + 8)] = 2 e^- 4t - e^- 2t