10.45, §2 Solution

y'' - 2 y' + y = 3 e^2x

Homogeneous equation:

Characteristic polynomial:

has roots :General solution:

y_h = C₁ e^x + C₂ x e^x

Particular solution:

y_p'' - 2 y_p' + y_p = 3 e^2x

Guessing y_p = C e^2x produces

so C=3 and y_p = 3 e^2x.

Total solution:

y = 3 e^2x + C₁ e^x + C₂ x e^x