second order differential equation

the trick

Here is a pretty ubiquitous trick to solve differential equations of the second order differential equations. It is used to change a second order differential equation to a First-Order Differential Equations.

If you have a differential equation of the shape:

\begin{equation} x^{’’} = f(x,x’) \end{equation}

that, the second derivative is strictly a function between the first derivative value and the current value.

We are going to define a notation \(x’ = v\), which makes sense.

So, we will describe:

\begin{equation} x^{’’} = \dv{v}{t} = \dv{v}{x} \dv{x}{t} = v\dv{v}{x} \end{equation}

So therefore, we have:

\begin{equation} x^{’’} = v\dv{v}{x} = f(x,v) \end{equation}

So turns out, the original input \(t\) is, given a specific equation above, we have no need to know it.

To actually go about solving it, see solving homogeneous higher-order differential equations.