Solving a nonlinear second order ODE initial value problem

Question

Consider the following problem. \begin{cases} y''+\frac{1}{x}y'-\frac{1}{x^2}y+(1-y^2)y=0 & \text{for }x>0,\\ y(0)=0, \end{cases} where $y=y(x)\in\mathbb R$.

1. Use the method of Frobenius to solve it.
2. Prove that there exists a solution $y$ which satisfies $y(x)>0$ for $x>0$ and $\lim_{x\to\infty}y(x)=1$.
3. Prove that there exists a solution $y$ which satisfies that $y$ has infinitely many zeros.

We are told to solve this problem; however, get stuck even at the first step. We have never encounter this system, and failed to use the method of Frobenius to solve it.