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:

$(1): \quad$ The values of the function itself
$(2): \quad$ Its derivatives of various orders.


Order

The order of a differential equation is defined as being the order 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 a differential equation which has exactly one independent variable.

All the derivatives occurring in it are therefore ordinary.


The general ODE of order $n$ is:

$f \left({x, y, \dfrac {\mathrm d x} {\mathrm d y}, \dfrac {\mathrm d^2 x} {\mathrm d y^2}, \ldots, \dfrac {\mathrm d^n x} {\mathrm d y^n}}\right) = 0$

or, using the prime notation:

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


Partial Differential Equation

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

The derivatives occurring in it are therefore partial.


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 a differential equation where all 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 a differential equation 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

A differential equation is autonomous if none of the derivatives depend on the independent variable.

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

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


System of Differential Equations

A system of differential equations is a set of simultaneous differential equations.

The solutions for each of the differential equations are in general expected to be consistent.


Explicit and Implicit

Explicit System

A differential equation is called explicit if it can be written in the form:

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


Implicit

A differential equation that is not explicit is referred to as implicit.


Also see

  • Results about differential equations can be found here.


Historical Note

Much of the theory of differential equations was established by Leonhard Paul Euler.

The first existence proof for the solutions of a differential equation was provided by Augustin Louis Cauchy.


Sources