4.5 ode of the form \(y^{\prime \prime }+a_{2}\left ( x,y\right ) \left ( y^{\prime }\right ) ^{2}+a_{1}\left ( x,y\right ) y^{\prime }+a_{0}\left ( x,y\right ) =0\)

From paper (On first integrals of second-order ordinary differential equations by Romero et all), this is called class B. The first integral is

\[ \frac {d}{dx}R=C\left ( x\right ) +\frac {1}{A\left ( x,y\right ) y^{\prime }+B\left ( x,y\right ) }\]

where \(C_{y}=0\). Another class of ode’s is called class \(A\) with first integral

\[ \frac {d}{dx}R=\frac {1}{A\left ( x,y\right ) y^{\prime }+B\left ( x,y\right ) }\]

This is subset of class B.