2.20.3.4.6 Examples for section 3-1-3 (b) Relations between the coefficients, subcase (iii)

2.20.3.4.6.1 Algorithm Assuming particular solution of \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\) is \(y_{1}\) where

\begin{equation} 2f_{2}y_{1}=X\left ( x\right ) -f_{1} \tag {1}\end{equation}

Where \(X\left ( x\right ) \) satisfies

\begin{equation} f_{0}=f_{2}y_{1}^{2}-Xy_{1}+y_{1}^{\prime } \tag {2}\end{equation}

Murphy book suggests to try

\begin{align*} X & =0\\ X & =-\frac {f_{2}^{\prime }}{f_{2}}\\ X & =f_{1}-2\sqrt {f_{0}f_{2}}\end{align*}

For each such case of \(X\), we end up with \(y_{1}\) from (1), which now we check if it satisfies (2) or not. If it does, then \(y_{1}\) can be used to solve the Riccati ode