First-order ODE

Description

First-order ODEs are typically non-linear equations of first order and implicitly of dimension one.

Note: This page serves as aMoC

Definition

An Ordinary differential equation is called first order (and dimension one) if it is of the form $x^{\prime }=f(t,x)$ where $f:G\subseteq {\mathbb{R}}^2\to \mathbb{R}$.

There are many different ways of solving, they are all described in examples.

Properties

Tip

Examples

Linear first-order equation

These are equation of the form: $x^{\prime }=f(t)x+g(t)$ The solution of linear first-order equations is described in Linear ODE.

Separation of variables

An equation of the form $x^{\prime }=f(t)g(x)$ can be solved using Separation of variables

Exact ODEs and almost exact ODEs

Exact differential equations are Implizite Differentialgleichung but technically first order. These are of the form $f(t,x)+g(t,x)x^{\prime }=0$ with a potential function attached. They can be solved by studying the Level sets of the Gradient of the potential function. Some other ODEs can be made exact by multiplying an integrating factor.

Change of variables

The following examples can be solved by Change of Variables. More at the page:

• Equations equivariant under scaling $f(t,x)=f(\sigma t,\sigma x)$
• Affine-linear transformation. Substituting a affine-linear map can make some stuff easier
• Bernoulli equations: An equation of the form $x^{\prime }+f(t)x=g(t)x^n$.

Tools

Bound solutions

We use solutions to different Initial value problem to bound one solution, we want to understand.