Chapter 17.2, Problem 55E

To determine

To verify: Whether the function $y_p=2e^x\cos x$ is a particular solution to the specified non homogeneous $\frac{1}{2}{y}^{″}+y^{\prime }+y=4e^x\left(\cos x-\sin x\right)$ and then find the general solution and evaluate its arbitrary constants to find the unique solution satisfying the equation and the initial conditions are $y\left(0\right)=0,y^{\prime }\left(0\right)=1$.