>>210
>Solving Riccati equation using simple integration is not possible unless one at least one particular solution is known
I think this is somewhat misleading. There are plenty of techniques available to handle nonlinear differential equations.
I think a helpful comment here is that it is often useful to use a linearising substitution like \begin{math}y=\psi'/\psi\end{math} which in some cases will reduce a Ricatti equation to a linear equation for \begin{math}\psi\end{math}. 
Without loss of generality, you can always reduce Ricatti equations to the form where the coefficient of \begin{math}y^2\end{math} is 1, (substitute for \begin{math}y\mapsto Py\end{math}) and then you can apply the substitution above. This linearises the equation.
This is an essential idea to know when handling the Ricatti equation and nonlinear differential equations in general.