Working backwords from solution to DE

[Panphobia]

Homework Statement

if y = c₁*e^t*sin(√3t)+c₂*e^t*cos(√3t) + sintcost

The Attempt at a Solution

I have already figured out that (2±√-12)/2 so b = -2 and -12 = 4-4c so c = 4 and a = 1
and that means that the associated homogeneous DE is y'' -2y' + 4y = 0, but I don't know what to do about the sintcost to get then g(t) in y'' -2y' + 4y = g(t), I have tried looking at different methods of solving 2nd order differential equations and tried working backwards but I can't get anything going. What would a good first step be?

 

[pasmith]

Homework Statement

if y = c₁*e^t*sin(√3t)+c₂*e^t*cos(√3t) + sintcost

The Attempt at a Solution

I have already figured out that (2±√-12)/2 so b = -2 and -12 = 4-4c so c = 4 and a = 1 and that means that the associated homogeneous DE is y'' -2y' + 4y = 0,but I don't know what to do about the sintcost to get then g(t) in y'' -2y' + 4y = g(t), I have tried looking at different methods of solving 2nd order differential equations and tried working backwards but I can't get anything going. What would a good first step be?

\sin t \cos t = \frac12 \sin(2t).

 

[Panphobia]

Yea I got the answer now, thanks a lot!