Here's the question: (b) Show that $y_1$ is a solution of the DE $y'+p(t)y=0$. (It is called a homogeneous DE, whenever $g(t) = 0$.)
This is the second part to this problem: (a) Show that the solution of the linear equation y'+p(t)y=g(t) can be written in the form $y=cy_1(t)+Y(t)$, where c is an arbitrary constant. Identify the functions $y_1(t)$ and $Y(t)$. I found : $$y_1 =\frac 1 {μ(t)} \text{ and } Y= \frac {\intμ(s)g(s)ds}{\mu(t)}$$.
I tried solving the homogeneous DE by separating the variable and I got to $\ln(y)=\int p(t)dt$... I'm not sure where to go from here to get $y=y_1$. Any help is appreciated!