When a nonlinear econd order ode is missing \(x\) then use the substitution \(y^{\prime }\left ( x\right ) =p\left ( y\right ) \). Example is \(yy^{\prime \prime }-\left ( y^{\prime }\right ) ^{2}=1\).
When a nonlinear econd order ode is missing \(y\) use the substitution \(y^{\prime }\left ( x\right ) =p\left ( x\right ) \). Example \(y^{\prime \prime }\left ( x\right ) =\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\) or or \(y^{\prime \prime }=\left ( y^{\prime }\right ) ^{2}\cos x\).
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 \(p\). These methods are meant to be used only when the second order ode is nonlinear.