When a nonlinear second order ode is missing \(x\) then make everything as \(\frac {du}{dy}\) using the substitution \(u=y^{\prime },y^{\prime \prime }=u\frac {du}{dy},y^{\prime \prime \prime }=u^{2}\frac {d^{2}u}{dy^{2}}+u\left ( \frac {du}{dy}\right ) ^{2}\) and so on. Example is \(yy^{\prime \prime }-\left ( y^{\prime }\right ) ^{2}=1\).
When a nonlinear second order ode is missing \(y\) then make everything \(\frac {du}{dx}\) using the substitution \(u=y^{\prime },y^{\prime \prime }=\frac {du}{dx},y^{\prime \prime \prime }=\frac {d^{2}u}{dx^{2}}\) and so on. Example \(y^{\prime \prime }\left ( x\right ) =\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\) or or \(y^{\prime \prime }=\left ( y^{\prime }\right ) ^{2}\cos x\). Notice that we start with the same substitution which is \(y^{\prime }=u\). See examples below.
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 becomes \(u\) which is then easily solved for. These methods are meant to be used only when the second order ode is nonlinear.
If the ode is missing both \(x\) and \(y\) then either method will work.