Download - Engineering Mathematics Topic 1
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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=
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-
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