TITLE: Prove that there is no solution in terms of elementary functions of $y'=x^2+y^2$
QUESTION [5 upvotes]: I have seen before that
$$y'=x^2+y^2$$
Has no solution in terms of elementary functions. I typed it in on Wolfram Alpha (here) and got a giant solution with a bunch of Bessel functions that I did not understand. This equation is almost like a Ricatti equation, but not exactly, so there may be some insight to be gained there. Again, my question is:
How do you prove that this differential equation has no solution in terms of elementary functions?

REPLY [5 votes]: It is a particular case of the Riccati equation. 
Set $\displaystyle y=-\frac{f'}{f}$, then $\displaystyle y'=-\frac{f''}{f}+\frac{f'^2}{f^2}$, so that the equation for $f$ is given by
$$f''+x^2f=0.$$
This is linear and solvable in terms of Bessel functions:
$$f(x)=C_+\sqrt{x}\,J_{\frac14}\left(\frac{x^2}{2}\right)+C_-\sqrt{x}\,J_{-\frac14}\left(\frac{x^2}{2}\right).$$
The answer given by Wolfram Alpha is nothing but the logarithmic derivative of this formula. Note that only the ratio $C_{+}/C_-$ will appear in the expression for $y$, providing the integration constant for the initial 1st order equation.

Bessel functions $J_{\pm\frac14}(z)$ can also be expressed in terms of parabolic cylinder functions as indicated here, but do not simplify further.