4.2 Exact nonlinear second order ode \(F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =0\) (Approach 2)

This method is based on paper "Exactness of Second Order Ordinary Differential Equations and Integrating Factors", by AlAhmad, M. Al-Jararha and H. Almefleh which now I have full implementation for. We start with the ode in the form

\begin{equation} a_{2}\left ( x,y,y^{\prime }\right ) y^{\prime \prime }+a_{1}\left ( x,y,y^{\prime }\right ) y^{\prime }+a_{0}\left ( x,y,y^{\prime }\right ) =0 \tag {1}\end{equation}

Then, we first verify the ode is exact using the conditions

\begin{align} \frac {\partial a_{2}}{\partial y} & =\frac {\partial a_{1}}{\partial y^{\prime }}\nonumber \\ \frac {\partial a_{2}}{\partial x} & =\frac {\partial a_{0}}{\partial y^{\prime }}\tag {2}\\ \frac {\partial a_{1}}{\partial x} & =\frac {\partial a_{0}}{\partial y}\nonumber \end{align}

If the above are satisfied, then next we generate a first order ode using

\begin{equation} \int _{x_{0}}^{x}a_{0}\left ( \alpha ,y,y^{\prime }\right ) d\alpha +\int _{y_{0}}^{y}a_{1}\left ( x_{0},\beta ,y^{\prime }\right ) d\beta +\int _{y_{0}^{\prime }}^{y^{\prime }}a_{2}\left ( x_{0},y_{0},\gamma \right ) d\gamma =0 \tag {3}\end{equation}

If we are not given initial conditions for the original ode, then the above is replaced by

\begin{equation} \int _{0}^{x}a_{0}\left ( \alpha ,y,y^{\prime }\right ) d\alpha +\int _{0}^{y}a_{1}\left ( 0,\beta ,y^{\prime }\right ) d\beta +\int _{0}^{y^{\prime }}a_{2}\left ( 0,0,\gamma \right ) d\gamma =c_{1} \tag {4}\end{equation}

Next, we solve the the above first order ode.  Examples below make this method more clear. Notice that when matching our equation against the template (1), it is possible to obtain different possible matches and hence different possible \(a_{0},a_{1},a_{2}\) depending on how the match is done. We should only pick one that satisfy the exactness conditions and use that match. See example 4 below for such an example to illustrate what this means.