differential equation

differential equation

A

differential equation

is an equation involving an unknown

function

of one or more variables, its

derivatives

and the
variables.
This type of equations comes up often in many different branches of
mathematics. They are also especially important in many problems in
physics and engineering.

There are many types of differential equations. An

ordinary differential
equation

(ODE) is a differential equation where the unknown function depends
on a single variable. A general ODE has the form

F

⁢

(

x

,

f

⁢

(

x

)

,

f

′

⁢

(

x

)

,

…

,

f

(

n

)

⁢

(

x

)

)

=

0

,

(1)

where the unknown

f

is usually understood to be a real or complex valued function of

x

, and

x

is usually understood to be either a real or complex
variable.
The
of a differential equation is the order of the highest derivative appearing
in Eq. (

1

). In this case, assuming that

F

depends
nontrivially on

f

(

n

)

⁢

(

x

)

, the equation is of

n

th order.

If a differential equation is satisfied by a function which
identically

vanishes

(i.e.

f

⁢

(

x

)

=

0

for each

x

in the
of interest),
then the equation is said to be

homogeneous

. Otherwise it is said to be

nonhomogeneous

(or

inhomogeneous

). Many differential equations
can be expressed in the form

L

⁢

[

f

]

=

g

⁢

(

x

)

,

where

L

is a

differential
operator

(with

g

⁢

(

x

)

=

0

for the homogeneous case). If the

operator

L

is
linear in

f

, then the equation is said to be a

linear

ODE and
otherwise

nonlinear

.

Other types of differential equations involve more complicated involving the unknown function.
A

partial differential equation

(PDE) is a differential equation
where the unknown function depends on more than one variable. In a

delay differential equation

(DDE), the unknown function depends on the state of the system at some instant in the past.

Solving differential equations is a difficult task. Three major types of approaches are possible:

•

Exact

methods are generally to equations of low order and/or to linear systems.

•

Qualitative

methods do not give explicit for the solutions, but provide pertaining to the

asymptotic

behavior of the system.

•

Finally,

numerical

methods allow to construct approximated solutions.

Examples

A common example of an ODE is the equation for

simple harmonic motion

d

2

⁢

u

d

⁢

x

2

+

k

⁢

u

=

0

.

This equation is of

second order

. It can be transformed into a system of two first order differential equations by introducing a variable

v

=

d

⁢

u

/

d

⁢

x

. Indeed, we then have

d

⁢

v

d

⁢

x

=

-

k

⁢

u

d

⁢

u

d

⁢

x

=

v

.

A common example of a PDE is the

wave equation

in three dimensions

∂

2

⁡

u

∂

⁡

x

2

+

∂

2

⁡

u

∂

⁡

y

2

+

∂

2

⁡

u

∂

⁡

z

2

=

c

2

⁢

∂

2

⁡

u

∂

⁡

t

2

Title

differential equation

Canonical name

DifferentialEquation

Date of creation

2013-03-22 12:41:22

Last modified on

2013-03-22 12:41:22

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 35-00

Classification

msc 34-00

Related topic

HeatEquation

Related topic

MethodOfUndeterminedCoefficients

Related topic

ExampleOfUniversalStructure

Related topic

CauchyInitialValueProblem

Related topic

Equation

Related topic

MaxwellsEquations

Defines

ordinary differential equation

Defines

ODE

Defines

partial differential equation

Defines

PDE

Defines

homogeneous

Defines

nonhomogeneous

Defines

inhomogeneous

Defines

linear differential equation

Defines

nonlinear differential equation

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