ラプラス変換を使って次の微分方程式を解く。
y" - 2y' - 3y = 0, y(0) = 1, y'(0) = 5
fのラプラス変換をLfとして
Ly = Y
Ly' = sY - y(0) = sY - 1
Ly" = s2 Y - sy(0) - y'(0) = s2 Y - s - 5
L1 = 1/s
これらを与式に代入
(s2 Y - s - 5) - 2(sY - 1) - 3Y = 1/s
(s2 - 2s - 3)Y = s + 3 + 1/s
Y = (s2 + 3s + 1)/[s(s + 1)(s - 3)] = [s(s + 1) + (s + 1) + s]/[s(s + 1)(s - 3)] = 1/(s - 3) + 1/[s(s - 3)] + 1/[(s + 1)(s - 3)] = 1/(s - 3) + [s - (s - 3)]/[3s(s - 3)] + [(s + 1) - (s - 3)]/[4(s + 1)(s - 3)] = 1/(s - 3) + 1/[3(s - 3)] - 1/(3s) + 1/[4(s - 3)] - 1/[4(s + 1)] = 19/12(s - 3) - 1/3s - 1/4(s+1)
逆ラプラス変換により
y = (19/12)exp(3t) - (1/3)exp(t) - (1/4)exp(-t)