I am aware that the solution to a homogeneous first order differential equation of the form $\frac{dy}{dx}=p(x)y$ can be obtained by simply by rearranging to:

$\frac{dy}{y}=p(x)dx$

Then it is simply a question of integrating both sides and the answer is straightforward. Now what would happen if RHS had a constant, how a can find a particular solution to this case: $\frac{dy}{dx}=p(x)y+C$

I know that the general solution would be the sum of the homogeneous equation and the particular solution

$\frac{dy}{y}=p(x)dx$

Then it is simply a question of integrating both sides and the answer is straightforward. Now what would happen if RHS had a constant, how a can find a particular solution to this case: $\frac{dy}{dx}=p(x)y+C$

I know that the general solution would be the sum of the homogeneous equation and the particular solution