A nonlinear ode with missing \(x\) has the form \begin {equation} y^{\prime \prime }+f\left ( y^{\prime }\right ) +g\left ( y\right ) =0 \tag {1} \end {equation} Where \(f\left ( y^{\prime }\right ) \) and \(g\left ( y\right ) \) can be nonlinear functions (one or both of them). If both are linear, then there is no need for this method to be used. For missing \(x\) the substitution \(y^{\prime }\left ( x\right ) =p\left ( y\right ) \) is used. For nonlinear ode with missing \(y\) which has the form \begin {equation} y^{\prime \prime }+F\left ( x\right ) f\left ( y^{\prime }\right ) +g\left ( x\right ) =0 \tag {2} \end {equation} Where \(F\left ( x\right ) ,g\left ( x\right ) \) are functions of \(x\) (or constants) and \(f\left ( y^{\prime }\right ) \) is nonlinear in \(y^{\prime }\). For this case the substitution \(y^{\prime }\left ( x\right ) =p\left ( x\right ) \) is used instead. The following gives examples of each method.
Both methods reduce the order of the ode by one resulting in first order ode where the dependent variable is \(p\) which is then easily solved for \(p\). This now results in another first order ode in \(y\) which is then easily solved.