Definition:Differential Equation

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Definition

A differential equation is a mathematical equation for an unknown function of one or several variables relating:


Order

The order of a differential equation is defined as being that of the highest order derivative that is present in the equation.


Ordinary and Partial Differential Equations

There are two types of differential equation:


Ordinary Differential Equation

An ordinary differential equation (abbreviated O.D.E. or ODE) is one which has only one independent variable.

All the derivatives occurring in it are therefore ordinary.


The general ODE of order $n$ is:

$\displaystyle f \left({x, y, \frac {dx} {dy}, \frac {d^2x} {dy^2}, \ldots, \frac {d^nx} {dy^n}}\right) = 0$

or, using the prime notation:

$f \left({x, y, y^{\prime}, y^{\prime \prime}, \ldots, y^{\left({n}\right)}}\right) = 0$


Partial Differential Equation

A partial differential equation (abbreviated P.D.E. or PDE) is one which has more than one independent variable.

The derivatives occurring in it are therefore partial.


Mixed Differential Equation

A mixed differential equation is one in which both ordinary derivatives and partial derivatives occur.


Linear and Non-Linear

Differential equations can also be classified as to whether they are linear or non-linear.


Linear

A linear differential equation is one where any dependent variables and their derivatives appear to the first power.

Neither are products of dependent variables allowed.


Non-Linear

A non-linear differential equation is one which is not linear.


Solution

Let $\Phi$ be a differential equation.

Any function $\phi$ which satisfies $\Phi$ is known as a solution of $\Phi$.


Note that, in general, there may be more than one solution to a given differential equation.

On the other hand, there may be none at all.


Solution Set

The solution set (or the solution) of $\Phi$ is the set of all functions $\phi$ that satisfy $\Phi$.


Autonomous System

A differential equation or system of differential equations is called autonomous if none of the derivatives depend on the independent variable.

The $n$th order autonomous differential equation takes the form:

$y^{\left({n}\right)} = f \left({y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$


Explicit System

A differential equation or system of differential equations is called explicit if it can be written in the form:

$y^{\left({n}\right)} = f \left({x,y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$

An ODE that is not explicit is implicit.

In practice the vast majority of ODEs are explicit; since such systems can be reduced to a first order problem, the theory of ODEs is concerned mainly with first order problems.


Sources