First-order first-degree autonomous differential equation

Definition

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

Form of the differential equation

The differential equation is of the form:

$\frac{dx}{dt}=f(x)$

Solution method and formula

We convert the differential equation to an integration problem:

$\int \frac{dx}{f(x)}=\int dt$

and carry out the integrations on both sides. If $p$ is an antiderivative of $1/f$, the solution will be:

$p(x)=t+C$

with $C$ the freely varying parameter over $\mathbb{R}$: every particular value of $C$ gives a solution. To express $x$ as a function of $t$, we need to invert $p$. If we can do so, we'd have:

$x=p^{-1}(t+C)$

as the general solution.

In addition, there may be stationary solutions. These are solutions that correspond to constant functions $x=k$ that satisfy $f(k)=0$.

Related notions

• Separable differential equation: A slightly more general type of first-order differential equation.
• Second-order autonomous differential equation of degree one: Although such equations cannot always be solved, they can always be reduced to first-order differential equations.

Analysis

Starting at time zero with value one

Suppose we want to solve the initial value problem for the differential equation:

$\frac{dx}{dt}=f(x)$

subject to the initial condition that at $t=0$, $x=1$.

We consider various possibilities for $f$ a function that sends 1 and higher numbers to positive numbers, and make cases based on the growth rate of $f$:

Nature of $f$	Nature of $x$ in terms of $t$?
constant function $f(x):=c$	linear function $x:=1+ct$
$f(x):=x^{\gamma },0<\gamma <1$	$x=O(t^{1/(1-\gamma )})$ (i.e., it grows roughly like a power function of $t$)
linear function $f(x):=mx,m>0$	exponential function $x:=\exp (mt)$
linear times logarithmic, something like $x(\ln x+1)$	$x$ grows something like a doubly exponential function of $t$ (note: if we wanted growth between exponential and double exponential, we would need something like $x(\ln x+1{)}^{\gamma },0<\gamma <1$, and if we wanted triple exponential growth, we would multiply by a double logarithmic term)
$f(x):=x^{\gamma },\gamma >1$	$x$ grows so fast in terms of $t$ that it reaches $\mathrm{\infty }$ in finite time.