Solve the Differential equation by inspection

Question

(D^3 - 7D - 6)y = x^2 + e^x

Answer 1

y'''- 7y' - 6y = x² + e^x; r³ - 7r - 6 = 0; r³ + 1 - 7r - 7 = 0; (r + 1)(r² - r - 6) = 0; r = - 1, r = - 2; r = 3

y_H = Ae^-x + Be^-2x + Ce^3x; y_P = Mx² + Nx + L + Ke^x; y_P' = 2M x + N + Ke^x, y_P'' = 2M + Ke^x; y_P''' = Ke^x.

Ke^x - 14Mx - 7N - 7Ke^x - 6Mx² - 6Nx - 6L - 6Ke^x = x² + e^x; - 12K = 1; - 14M - 6N = 0; - 6M = 1; - 7N - 6L = 0

K = - 1/12, M = - 1/6; N = -7/3·(- 1/6) = 7/18; L = - 7/6·7/18 = - 49/108.

y = Ae^-x + Be^-x + Ce^3x - 1/6x² + 7/18x - 49/108 - 1/12e^x.