1.3.6.1 Algorithm

Given \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\), this section shows how to attempt to find a particular solution. This is probably the most important part, since if we can find a particular solution, then the Riccati ode is solved. But there is no algorithm which works all the time to find a particular solution, since if there is, then Riccati can now be solved in all cases. But it is not.  

The main algorithm shown by Murphy and Kamke books starts by assuming that \(y=\frac {u\left ( x\right ) }{f_{2}}\) and transforms to  new ode in \(u\) which is

\[ u^{\prime }=F\left ( x\right ) +G\left ( x\right ) u+u^{2}\]

Where

\begin{align*} F & =f_{0}f_{2}\\ G & =f_{1}+\frac {f_{2}^{\prime }}{f_{2}}\end{align*}

Only if \(F,G\) are polynomials in \(x\), then more progress can be made as shown in the flow chart. There are two main subcases. When \(G=0\) and when \(G\neq 0\). Examples below go over each subcase.  If \(F,G\) are not polynomials then no progress can be made. Guessing  a particular solution can be tried but guessing is not an algorithm. Note that even if \(f_{0},f_{1},f_{2}\) are polynomials, this does not necessarily implies that \(F,G\) will be polynomials.