Missing x examples

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 \(y^{\prime }=u\) then \(y^{\prime \prime }=\frac {du}{dx}=\frac {du}{dy}\frac {dy}{dx}=u\frac {du}{dy}\) and the ode becomes

\begin{equation} u^{\prime }u=f\left ( y,u\right ) \tag {2}\end{equation}

Which is now a ﬁrst order ode. If we can solve this for \(u\) then the solution to the original ode (1) is

\begin{align*} \frac {dy}{dx} & =u\left ( y\right ) \\ \int \frac {dy}{u\left ( y\right ) } & =x+c_{1}\end{align*}