# Analytic solution?

1. May 11, 2010

### Malmstrom

1. The problem statement, all variables and given/known data
Consider the following ODE
$$x^3y' - 2y + 2x = 0$$

2. Relevant equations
Prove that the ODE has no analytic solution in any neighborhood of $$x=0$$.

3. The attempt at a solution
Its general solution is $$Ce^{-\frac{1}{x^2}}+ e^{-\\frac{1}{x^2}} \int_{x_0}^x - \frac{2e^{\frac{1}{t^2}}}{t^2} dt$$ which is not continuous in $$x=0$$, but I don't think this is unique so I don't know if it helps. Should I try to substitute an arbitrary power series in the equation?