How to solve $x^2 \dfrac{d^2y}{dx^2} - 2y = x^2 + \frac{1}{x}$

Question

I have seen how to solve these kinds of equations when a first solution is given. But what about this case when no first solution is given?

Answer 1

Hints: this is a non-homogeneous Cauchy-Euler equation. Solve the homogeneous case by trying a solution $y(x)=x^m$ and determine two values of $m$ to obtain two linearly independent solutions. Then, use variation of parameters to find a particular solution.

Answer 2

HINT

Here I present you another way to approach it

\begin{align*} x^{2}y^{\prime\prime} - 2y & = (x^{2}y^{\prime\prime} + 2xy^{\prime}) - (2xy^{\prime} + 2y) = (x^{2}y^{\prime})^{\prime} - 2(xy)^{\prime} = x^{2} + \frac{1}{x} \end{align*}

Can you take it from here?