Terminology

Ordinary differential equations: Introduction

Terminology

Order and degree of an ODE 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 \(y\).

Example 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? \[\frac{\dd^3y}{\dd t^3}\Bigl/\frac{\dd y}{\dd t}=\left(\frac{\dd ^2y}{\dd t^2}\right)^{\!2}\]

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 \[\frac{\dd^3y}{\dd t^3}-\frac{\dd y}{\dd t}\cdot \left(\frac{\dd^2y}{\dd t^2}\right)^{\!2}=0\] 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

Types of differential equations 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.

Examples 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 \[a_n(t)\cdot y^{(n)}+\cdots + a_2(t)\cdot y'' +a_1(t)\cdot y'+a_0(t)\cdot y + b(t)=0\]