In my class we learned how to solve DEs using the variation of parameters when the coefficients are constant, we use undetermined coefficients to get two homogeneous solutions then apply the method of variation of parameters to get a particular solution. But I was never taught how to use method of variation of parameters when the coefficients are non-constant. For example
Has solutions $y_1=x$ and $y_2=x^3$.
I found this example online but even it doesn't show how these homogenous solutions were found... But it uses them to find the particular solution.... Which I'm fine with.
How do I find these solutions and is there a general strategy? Thanks!