1.3.4.4.1 Algorithm Given \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\), we look for \(a,b\) constants not both \(a,b\) zero such that

\[ a^{2}f_{0}+abf_{1}+b^{2}f_{2}=0 \]

If we can find such \(a,b\), then there are three cases to consider.

1. \(a\neq 0\). In this case a particular solution is \(y_{1}=\frac {b}{a}\). Hence  now we can solve the Riccati ode since a particular solution is known.
2. \(a=1,b=1\), then this means \(f_{0}+f_{1}+f_{2}=0\) where now a particular solution is \(y_{1}=1\), and now we can solve the Riccati ode since a particular solution is known
3. case \(a=0\) and \(b\neq 0\) can not show up, since this implies \(f_{2}=0\) and hence the ode is not Riccati to start with.