Answer
See below
Work Step by Step
$y''+2y'+y=50\sin 3x$
The complementary function is
$y_c(x)=c_1e^{-x}+c_2e^{-x}$
We consider the complex differential equation
$z''+2z'+z=50e^{3xi}$
An appropriate complex-valued trial solution for this differential equation is:
$z_p(x)=A_0e^{3xi}$
where $A_0$ is a complex constant. The derivatives of $z_p$ is
$z''_p(x)=(3i)^2A_0e^{3xi}\\
z'_p(x)=(3i)A_0e^{3xi}$
so that $z _p$ is a solution if and only if
$(3i)^2A_0e^{3xi}+2(3i)A_0e^{3xi}+A_0e^{3xi}=50e^{4xi}\\
-20A_0e^{4xi}=50e^{3xi}\\
A_0=\frac{50}{6i-8}$
A particular solution is:
$y_p(x)=Re_{\{z_p\}}=-4\sin 3x -3\cos 3x$
so that the general solution is
$y(x)=c_1e^{-x}+c_2e^{-x}-4\sin 3x -3\cos 3x$
