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.