Riccati differential equation

The Riccati differential equation, $y'=x+y^2$ is special equation. I know that how can I solve it, but my problem is that I don't have initial conditions, and I firstly need a particular solution. How to find the particular solution, please?

Solution 1:

$$y'=x+y^2$$ The change of function $y=-\frac{u'}{u}$ tramsforms the ODE to : $$u''+xu=0$$ This second order linear ODE is of the Bessel kind. $$u= c_1\sqrt{x} J_{1/3} (\frac{2}{3}x^{3/2}) +c_2\sqrt{x} J_{-1/3} (\frac{2}{3}x^{3/2}) $$ In order to compute $y=-\frac{u'}{u}$ , see the formula of the derivatives of the Bessel functions : http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/20/01/02/