Chapter 6.1, Problem 19E

In Problems $19-22$, a particular solution and a fundamental solution set are given for a nonhomogeneous equation and its corresponding homogeneous equation. (a) Find a general solution to the nonhomogeneous equation. (b) Find the solution that satisfies the specified initial conditions.

$y^{‴}+y^{″}+3y^{\prime }-5y=2+6x-5x^2$;

$y\left(0\right)=-1,y^{\prime }\left(0\right)=1,y^{″}\left(0\right)=-3$;

$y_p=x^2$; $\left\{e^x,e^{-x}\cos 2x,e^{-x}\sin 2x\right\}$