Putnam 1979 B4

Problem B-4 (a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$\left(3 x^{2}+x-1\right) y^{\prime \prime}-\left(9 x^{2}+9 x-2\right) y^{\prime}+(18 x+3) y=0$$

Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$\left(3 x^{2}+x-1\right) y^{\prime \prime}-\left(9 x^{2}+9 x-2\right) y^{\prime}+(18 x+3) y=6(6 x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1$. Find integers $a, b, c$ such that $(f(-2)-a)(f(2)-b)=c$.