---
title: "second order differential equation"
type: concept
related: [Second Order Differential Equations, Diffeq Intro]
source: https://www.jemoka.com/posts/kbhsecond_order_differential_equations/
confidence: high
status: active
---

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^{&rsquo;&rsquo;} = f(x,x&rsquo;) \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 \(x&rsquo; = v\), which makes sense.
So, we will describe:
\begin{equation} x^{&rsquo;&rsquo;} = \dv{v}{t} = \dv{v}{x} \dv{x}{t} = v\dv{v}{x} \end{equation}
So therefore, we have:
\begin{equation} x^{&rsquo;&rsquo;} = 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.
