Displacement x of a system

Question

Displacement x of a system satisfies $$ 3\ddot{x}+8\dot{x}+5x=43+2y-7\dot{y} $$ where $y=4\cos(t)$. If $x=0=$ $\dot{x}$ at $t=0$, find $x $and describe the motion that occurs at large times.

Thanks for all the help!

Answer 1

for the homogeneous part of the equation we get $$x(t)=e^{-5/3t}C_1+C_2e^{-t}$$ for the particular solution make the ansatz $$x_p(t)=A+B\cos(t)+C\sin(t)$$