I need help solving a differential equation

Question

Can someone please help me solve this differential equation.

$\frac{dy}{dx} =x^2 + y^2$.

Answer 1

$$\frac{dy}{dx}=x^2+y^2$$ This is a Ricati ODE. In the general case, the usual method to solve it is a change of function : $$y(x)=-\frac{f'(x)}{f(x)}\qquad\to\qquad f''+x^2f=0$$ The new ODE is a particular case of parabolic cylinder ODE which solutions are linear combinations of the parabolic cylinder functions : http://mathworld.wolfram.com/ParabolicCylinderDifferentialEquation.html

In this particular cases, the ODE can be transformed to an ODE of Bessel kind which general solution is : $$f(x)=c_1\sqrt{x}\:J_{1/4}\left(\frac{1}{2}x^2\right)+c_2\sqrt{x}\:J_{-1/4}\left(\frac{1}{2}x^2\right)$$ The derivative $f'(x)$ can be expressed in terms of Bessel functions in order to obtain the analytical result $y(x)=-\frac{f'(x)}{f(x)}$