Ordinary differential equation

d^2y/dt - 3dy/dt + 2y = 4e^(2t)
Integrate both sides wrt t :
dy/dt - 3y + 2ty = 2e^(2t) + C
5 - 3(-3) +2(0)(-3) = 2e^(2x0) (since y'(0)=5 y(0)=-3)
C = 12

Therefore,
dy/dt - 3y + 2ty = 2e^(2t) + 12
Integrate both sides wrt t again :
y - 3ty + yt^2 = e^(2t) + 12t + D
-3 - 3(0)(-3) + (-3)(0)^2 = e^(2x0) + 12(0) + D
D = -4

Therefore
y - 3ty + t^2y = e^(2t) +12t -4
y = [e^(2t) +12t -4]/(1-3t+t^2)