Solve the DE using reduction of order

Question

y" + 4y' + 8y = -t^2

Answer 1

r² + 4r + 8 = 0; r = - 2 ± 2i; y = e^-2t(Acos2t + Bsin2t);

y_P = Mt² + Nt + L; y_P' = 2Mt + N; y_P'' = 2M; 2M + 8Mt + 4N + 8Mt² + 8Nt + 8L = 0; 8M = -1; 8M + 8N = 0;

2M + 4N + 8L = 0; M = - 1/8; N = 1/8; L = - 1/32; y_P = -1/8t² + 1/8t + 1/32

y = e^-2t(Acos2t + Bsin2t) - 1/8t² + 1/8t + 1/32