When a nonlinear second order ode is missing \(x\) then we make everything as \(\frac {dp}{dy}\) using the substitution \(p=y^{\prime },y^{\prime \prime }=p\frac {dp}{dy},y^{\prime \prime \prime }=p^{2}\frac {d^{2}p}{dy^{2}}+p\left ( \frac {dp}{dy}\right ) ^{2}\) and so on. Example is \(yy^{\prime \prime }-\left ( y^{\prime }\right ) ^{2}=1\).
When a nonlinear second order ode is missing \(y\) then we make everything \(\frac {dp}{dx}\) using the substitution \(p=y^{\prime },y^{\prime \prime }=\frac {dp}{dx},y^{\prime \prime \prime }=\frac {d^{2}p}{dx^{2}}\) and so on. Examples are \(y^{\prime \prime }=\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\) and \(y^{\prime \prime }=\left ( y^{\prime }\right ) ^{2}\cos x\). Notice that we start with the same substitution which is \(y^{\prime }=p\). Some ode’s are missing \(x\) and also missing \(y\) such as \(y^{\prime \prime }=\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\) and for this any one of these methods work. 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.