Solution of $xy'+y+x^3y^3e^{-x^2}=0$

Question

Consider the following ode :

$$xy'+y+x^3y^3e^{-x^2}=0$$

I solved it as a Bernoulli equation and found :

$y(x)= \frac{\pm 1}{\sqrt{2x^2 \int e^{-x^2}}dx}$

I know I can write $\int e^{-x^2}dx$ using the error function(this is the solution given by Mathematica), but I was wondering if there is another way to write it. I thought about writing the Taylor series of $e^{-x^2}$ but I am not sure if I can swap the integral and the summation in order to evaluate the integral on this situation(indefinite integral). So my question is :

Is there another way to write $y(x)$ or is the error function the only option?

Answer 1

$$xy'+y+y^3x^3e^{-x^2}=0$$ $$\Rightarrow x\frac{y'}{y^3}+\frac{1}{y^2}+x^3e^{-x^2}=0$$ And this looks a bit like the Cauchy-Euler equation, maybe try a substitution for $y$?