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1 - CLASSIFICATION OF DIFFERENTIAL EQUATION (DE)

A solution of a differential equation is a relationship between the dependent n

independent variables.

Example:

Verify that xey = is a solution of the differential equation ydx

dy= .

Given xey = , thenxe

dx

dy=

Since: ydx

dy= therefore: xey = is a solution.

A function witharbitrary constant is ageneral solutionto a E.

A function withoutarbitrary constant is a partiular solutionto a E.

E to!ether with initial conditions is called an initial !alue pro"le#.

"he initial conditions are used to determine the values of arbitrary constants in the

!eneral solution

A stan$ar$ %or# o% a DE

#$#$#$...#$#$ %&&

&

& xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n =++++

'inear is characteri(ed by two properties)

"he dependent variable yand all its derivatives are of the first de!ree i.e.: the

power of each item involved is &

Each coefficient depends on only the independent variablex

An equation that is not linear is said to be nonlinear

Examples:

& * +lassification E &

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04)( =+ xdydxxy linear

02 =+ yyy linear

xey

dx

dyx

dx

yd=+ 5

3

3

linear

22 1)1( ydx

dyx +=+ non*linear

Or$inar& Di%%erential E'uationis an equation relatin! an unnown function or functions

to one or more of its derivatives.

Examples:

##$cos$#$

txdt

tdx=

dx

dyy

dx

yd=

-

-

Examples:

i# xey

dx

dy=+5

ii# 062

2

=+ ydx

dy

dx

yd

iii# yxdt

dy

dt

dx+=+ 2

artial Di%%erential E'uation contains several independent variables and partial

derivatives.

Examples:

-

- #,$#,$

du

utxd

dt

utdx=

-

-

-

-

-

udy

ud

dx

ud=+

Examples:

& * +lassification E -

nly one independent variable)t

nly one independent variable)x

"here are two independent variables)x is afunction oft andu

"here are two independent variables)u is afunction ofx andy

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i# 02

2

2

2

=

+

y

u

x

u

ii#t

u

t

u

x

u

=

2

2

2

2

2

iii# x

v

y

u

=

/n !eneral, a E will have many solutions $due to calculus that inte!ration introduces

arbitrary constants#

"he order of a E is the order of the hi!hest derivative that appears in the equation.

First Or$er ODE

Standard form for a &strder E which is linear can be written as follows:

#$#$ xryxpy =+

"he equation is linear in yand y ) wherepand rmay be any function ofx.

/f:

%#$ =xr then the equation is o#ogeneous

%#$ xr then the equation is non-o#ogeneous

Example of a &storder E:

012 xx

dx

dy=+

Seon$ Or$er ODE

Standard form for a -ndrder E which is linear can be written as follows, withp, qand r

may be any function ofx:

& * +lassification E 3

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#$#$#$ xryxqyxpy =+

and nonlinear if it cannot be written in this form

/f:

%#$ =xr then the equation is o#ogeneous

%#$ xr then the equation is non-o#ogeneous

Example of a -ndorder E:

xeydx

dy

dx

yd=+ 45

2

2

ntOr$er ODE

"he nthorder E is linear if it can be written as follows) with p, q and rmay be any

function ofx:

#$#$#$...#$ %&&

& xryxpyxpyxpy n

nn

=+++

and nonlinear if it cannot be written in this form

/f:

%#$ =xr then the equation is o#ogeneous

%#$ xr then the equation is non-o#ogeneous

Examples:

a#%=+ ydxxdy

'inear 4 &st rder $o#ogeneous#

b# %-55 =+ yyy 'inear 4 -nd rder $o#ogeneous#

& * +lassification E 1

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c#2

22

3

33

dx

ydx

dx

ydx = 'inear 4 3rd rder $non-o#ogeneous#

d# xyyy = 555 - 'inear 4 -nd rder $non-o#ogeneous#

e# %-3

3

= ydx

yd6onlinear 4 3rd rder $o#ogeneous#

& * +lassification E 2