Second-order first-degree autonomous differential equation

Definition

Following the convention for autonomous differential equations, we denote the dependent variable by $x$ and the independent variable by $t$.

Form of the differential equation

A (one-dimensional) second-order autonomous differential equation is a differential equation of the form:

$\frac{d^{2}x}{d{t}^2}=f(x)$

Solution method and formula

We set a variable $v=dx/dt$ Then, we can rewrite $dt=(dx)/v$. In particular, $d^{2}x/d{t}^2=(d/dt)(dx/dt)=dv/dt=dv/((dx)/v)=vdv/dx$. Plug this in:

$v\frac{dv}{dx}=f(x)$

This is now a separable differential equation relating $x$ and $v$. Integrate and obtain:

$\int vdv=\int f(x)dx$

We thus get:

$\frac{v^2}{2}=\int f(x)dx$

In particular, if $F$ is an antiderivative for $f$, then we get:

$v=\sqrt{2F(x)+C_1}$

where $C_1\in \mathbb{R}$ is a parameter. Each choice of $C_1$ gives a different solution.

Plug this back in and get:

$\frac{dx}{dt}=\sqrt{2F(x)+C_1}$

This is a first-order autonomous differential equation, and in particular a separable differential equation. Rearrange and get:

$\int \frac{dx}{\sqrt{2F(x)+C_1}}=\int dt$

An additional constant, $C_2$, arises from this indefinite integration. The upshot is that the general solution relates $x$ to $t$ and has two parameters $C_1,C_2$, as we might expect from the degree of the equation.