ode internal name "exact_nonlinear_second_order_ode_using_first_integral"
3.4.2.3.1 Introduction Not implemented yet. This uses point Lie symmetry.
The above section showed how to solve the nonlinear ode \(F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =0\) once it is determined it is exact as is, which is by finding the first integral \(R\) directly without finding an integrating factor first. This below gives few ode forms with the corresponding first integral \(R\) to use and how to find \(R\). These are collected from few papers I am studying now.
3.4.2.3.2 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.