Solving $\frac{dy}{dx}=1+(a_mx^m+a_{m-1}x^{m-1}+...+a_0)y^2$

Parametrize $y=\frac{u}{u'}$ to get the equation $$ u''(x)+P(x)u(x)=0. $$ There are some special cases of this equation that have solutions in special functions, like Airy or Bessel functions, but in general it is not the case that this can be reduced to named and classified equations.