How to solve $(x^2+1)y''-2xy'+2y=6(x^2+1)^2$, $y_1=x$
If you know the solution of homogeneous equation, to solve nonhomogeneous equation you only need to find one particular solution.
One can see that on the left you have a polynomial of 4-th degree. If $y(x)$ is a polynomial too, then all three terms on the right have the same degree. So it makes sense to search for the solution within polynomials of 4th degree.
If you know that a polynomial has form $$ y_n(x) = ax^4+bx^3+cx^2+dx +e, $$ can you find at least one set of $a,b,c,d,e$?