In this problem you will solve the non-homogeneous differential equation $$y''+ 12 y'+ 32 y =\sin\!\left(e^{4x}\right)$$

(1) Let $C_1$ and $C_2$ be arbitrary constants. The general solution to the related homogeneous differential equation $y''+ 12 y'+ 32 y =0$ is the function $y_h(x)=C_1\ y_1(x)+C_2\ y_2(x)=C_1$ $+C_2$.

NOTE: The order in which you enter the answers is important; that is, $C_1 f(x) + C_2 g(x) \neq C_1 g(x) + C_2 f(x)$.

(2) The particular solution $y_p(x)$ to the differential equation $$y''+ 12 y'+ 32 y =\sin\!\left(e^{4x}\right)$$ is of the form $y_p(x)=y_1(x)\ u_1(x)+y_2(x)\ u_2(x)$ where $u_1'(x)=$ and $u_2'(x)=$ .

(3) It follows that $u_1(x)=$ and $u_2(x)=$ ; thus $y_p(x)=$ .

(4) The most general solution to the non-homogeneous differential equation $y''+ 12 y'+ 32 y =\sin\!\left(e^{4x}\right)$ is
$y=C_1$ $+C_2$$+$.