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:

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

Solution method and formula

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

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

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

$\int v\,dv=\int f(x)\,dx$

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.