10.47, §2 Solution

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

Homogeneous equation:

y_h = C₁ e^x + C₂ x e^x

Particular solution:

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

Particular solutions y_p = e^x or y_p = x e^x don't work. Try

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

so C=1.5.

Total solution:

y = 1.5 x² e^x + C₁ e^x + C₂ x e^x