-- inverse Laplace transform for 1 / (s^2 + a s + b) -- from Wolfram T = -(exp(t (-1/2 sqrt(a^2 - 4 b) - a/2)) - exp(t (1/2 sqrt(a^2 - 4 b) - a/2)))/sqrt(a^2 - 4 b) T = eval(T, a^2 - 4 b, K^2) T1 = exp(1/2 K t) / K T2 = -exp(-1/2 K t) / K check(T == (T1 + T2) exp(-1/2 a t)) -- factor out i k = sqrt(4 b - a^2) K = i k check(T1 + T2 == 2 sin(1/2 k t) / k) f = 2 sin(1/2 k t) / k exp(-1/2 a t) check(T == f) F = 1 / (s^2 + a s + b) F "Inverse transform is" f Run