Answer
See below
Work Step by Step
$y''+16y=34e^{x}+16\cos 4x -8sin 4x$
We consider the complex differential equation
$z''+16z=34e^x+16^{4ix}-8e^{4ix}$
An appropriate complex-valued trial solution for this differential equation is:
$z_{p1}(x)=A_0e^x\\
z_{p2}=B_0xe^{4ix}\\
z_{p3}=C_0xe^{4ix}$
where $A_0,B_0,C_0$ are complex constants.
so that $z _p$ is a solution if and only if
$17A_0e^x+8iB_0e^{4ix}+8iC_0e^{4ix}=34e^x+16^{4ix}-8e^{4ix}\\
\rightarrow A_0=2\\B_0=-2i \\ C_0=i$
Hence, $z_{p1}=2e^x=2x \sin 4x-2ix\cos 4\\
z_{p2}=-2ie^{4ix}=34e^x+16^{4ix}-8e^{4ix}\\
z_{p3}=ixe^{4ix}=-x\sin 4x+ix \cos 4x$
A particular solution is:
$y_p(x)=Re_{\{z_p\}}=2e^{x}+2x\sin 4x+x\cos 4x$
