Terminology

Ordinary differential equations: Introduction

Terminology

The general form of an ordinary differential equation for a function $y$ of a single variable $t$ on a certain interval is $\varphi(t,y,y',y'',\ldots)=0$ where $\varphi$ is a $\varphi$ of several variables.

The order of this ordinary differential equation is the order of the highest derivative of $y$ present in $\varphi$ .
If the function $\varphi$ is a polynomial function in each of the derivatives of $y$ , then the degree of this ordinary differential equation is equal to the degree of $\varphi$ as a polynomial in the highest derivative.

If there are no derivatives in the equation, the order is equal to zero.

Instead of an ODE of order 1, we also call it a first-order differential equation. In general, we also call an ODE of order $n$ an $n$ -th order differential equation.

In determining the degree of an ODE of order $n$ we require that $\varphi(t,y,y',\ldots,y^{(n)})$ is a polynomial in $y'$ and higher derivatives, but we do not require that it is a polynomial in $t$ or in $y$ .

The differential equation $y''+\omega^2\cdot y=0$ with general solution $y(t)=C_1\sin(\omega\cdot t)+C_2\cos(\omega\cdot t)$ where $C_1$ and $C_2$ are constants, is a second-order differential equation, because the highest derivative in the differential equation is $y''$ . The degree of this ODE is equal to $1$ because the term with the highest derivative is equal to $y''$ .

What is the order and the degree of the following differential equation? $\frac{\dd^3y}{\dd t^3}\Bigl/\frac{\dd ^2y}{\dd t^2}=\left(\frac{\dd y}{\dd t}\right)^{\!2}$

The order of the ODE is $3$ .
The degree of the ODE is $1$ .

After all, the highest derivative in the ODE is $\frac{\dd^3y}{\dd t^3}$ and this means that the order of the ODE is equal to $3$ .
Rewrite the ODE as a polynomial equation: This is $\frac{d^3y}{dt^3}-\frac{d^2y}{dt^2}\cdot\left(\frac{dy}{\dd t}\right)^{\!2}=0$ The term in the ODE with the highest degree as polynomial equation in $\frac{\dd^3y}{\dd t^3}$ is herein $\frac{\dd^3y}{\dd t^3}$ , and this means that the degree of the ODE is equal to $1$ .

New example

The differential equation $\varphi(t,y,y',y'',\ldots)=0$ is called autonomous or time-invariant when the function $\varphi$ does not depend on the independent variable $t$ . The differential equation is called linear when the function $\varphi$ leads to an expression in which $y$ and its derivatives appear separately and not as a power of exponent different from 1, or as a product of each other. Otherwise, the ODE is non-linear.

The first order ODE $\frac{dy}{dt}=t\,y$ can be written as $\varphi(t,y,y')=0$ with $\varphi(u,v,w)=u\cdot v-w$ It is a differential equation of order 1 and degree 1. We also call it a first-order ODE of degree one.

The ODEs $y'-y^2=0\quad\text{and}\quad y'-y-1=0$ are examples of autonomous first-order differential equations of degree 1.

The ODE $(y'')^3-y^2=0$ is an example of an autonomous second-order differential equation of degree 3.

The ODE $(y'')^3-y\cdot y'=0$ is an example of a non-linear second-order differential equation of degree 3.

An autonomous differential equation of order $n$ $\left(\text{with }n\text{-th derivative }y^{(n)}\right)$ can often be written in the form $y^{(n)}=\Phi\bigl(y,y',y'',\ldots, y^{(n-1)}\bigr)$ with some function $\Phi$ .

A linear ODE of order $n$ can be written in the form $a_n(t)\cdot y^{(n)}+\cdots + a_2(t)\cdot y'' +a_1(t)\cdot y'+a_0(t)\cdot y + b(t)=0$