ode internal name "second_order_ode_missing_x"
Given \begin {equation} y^{\prime \prime }=f\left ( y,y^{\prime }\right ) \tag {1} \end {equation} Let \(p=y^{\prime }\) then \(y^{\prime \prime }=\frac {dp}{dx}=\frac {dp}{dy}\frac {dy}{dx}=\frac {dp}{dy}p=pp^{\prime }\) and the ode becomes\begin {equation} pp^{\prime }=f\left ( y,p\right ) \tag {2} \end {equation} Which is now a first order ode. If we can solve this for \(p\) then the solution to the original ode (1) is
\begin {align*} \frac {dy}{dx} & =p\left ( y\right ) \\ \int \frac {dy}{p\left ( y\right ) } & =x+c_{1} \end {align*}