ode internal name "higher_order_ODE_missing_x"
If the ode which is missing \(x\) then the substitution \(y^{\prime }=u,y^{\prime \prime }=u\frac {du}{dy},y^{\prime \prime \prime }=u^{2}\frac {d^{2}u}{dy^{2}}+u\left ( \frac {du}{dy}\right ) ^{2}\) and so on is used to reduced the order by one. This works for linear and nonlinear ode. Note if the higher order ode is missing both \(x\) and \(y\) at same time, then this substitution and the next one below also work.