Find the general solution of the problem.

Question

y'' + y' - 12y = 40t + 24t²

Note: Use t as the independent variable.

Use c₁ and c₂ as arbitrary constants

y(t) =

Answer 1

Characteristic equation: r² + r - 12 = 0; r = -4 or r = 3

General solution of homogenies solution: y_H = C₁e^-4t + C₂e^3t.

Particular solution of given ODE: y_P = at² + bt + c; y'_P = 2at + b; y''_P = 2a.

2a + 2at + b - 12at² - 12bt - 12c = 40t + 24 t²; 2a + b - 12c = 0; 2a - 12b = 40; -12a = 24, a = - 2; b = - 11/3

-4 -11/3 - 12c = 0; -12c = 23/3; c = - 23/36

y_p = - 2t² --11/3t - 23/36

General solution y(t) = C₁e^-4t + C₂e^3t - 2t² --11/3t - 23/36