Question
Use reduction of order to find a solution of the given nonhomogeneous equation. The indicated function $y_1(x)$ is a solution of the associated homogeneous equation. Determine a second solution of the homogeneous equation and a particular solution of the nonhomogeneous equation.
$$ y''-4y=2; y_1=e^{-2x} $$
I am very confused on their free use of "homogeneous" and "nonhomogeneous". This is how I am understanding the question:
"the given nonhomogeneous equation": $y''-4y=2$
"the associated homogeneous equation": ???
My teacher told me that a homogeneous equation in this context is one that equals zero, however there is none here. also, when I plug $y_1$ into $y''-4y=2$ it is not valid because you get $0=2$.
The homogeneous equation associated with $y''-4y=2$ is $y''-4y=0$. Once you know the solutions $y_1$, $y_2$ of the homogeneous equation, you can easily find the particular solution using the method of variation of parameters: $y(x)=u_1(x)y_1(x)+u_2(x)y_2(x)$. See here for more.