y'' - 2 y' + y = 3 e2x
Homogeneous equation:
Characteristic polynomial:
yh = C1 ex + C2 x ex
Particular solution:
yp'' - 2 yp' + yp = 3 e2x
Guessing yp = C e2x produces
Total solution:
y = 3 e2x + C1 ex + C2 x ex