Download - Lect. 1 IU EM301 Unit 1
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8/12/2019 Lect. 1 IU EM301 Unit 1
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Higher Order Linear Differential
EquationsPrepared By-Rootvesh Mehta
1Sci.& Hum.Dept. ,E.M.-319/07/2013
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Basic concepts
Definition- Differential Equation -- A differential equation
is an equation containing an unknownfunction and its derivatives. Examples are
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3
3
y
dxdy
dx yd
0 z z
x y
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Differential equations can beclassified in two parts
(1) Ordinary differential equations(2) Partial differential equations
An Ordinary differential equation is anequation which involves ordinary derivativesis an example of O.D.E
A partial differential equation is an equationwhich involves partial derivatives
Is an example of P.D.E
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3
y
dxdy
dx yd
0 z z
x y
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Order of Differential Equation
The order of the differential equation is theorder of the highest order derivative in thedifferential equation
Differential Equation ORDER
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32 xdxdy
1
09322
ydxdy
dx yd 2
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Degree of Differential Equation
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The degree of a differential equation is power of the highest orderderivative term in the differential equation.
Differential Equation Degree
0322
aydxdy
dx yd
0353
2
2
dxdy
dx yd
1
3
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Solution of a differential equation
Definition--The solution of a differentialequation is a function which satisfies givenequationtypes of solutions of differential equations1) General solution2) Particular Solution
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General Solution- The solution of differential equationis called general if the no. of arbitrary constants equals
to the order of differential equationParticular solution- If we assign particular value toarbitrary constant in general solution then it is calledparticular solution
Example - y=3x+c is solution of the 1st
order differentialequation ,here its a general solution
Now ,if we take c=5 in y=3 x+c then its a particularsolution.
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Solution of first order Lineardifferential equations
The solution of first order linear diff.eqn. canbe obtained as follows ,
Integrating factor is The General solution is
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Pdx
e
( . .) ( . .) y I F Q I F dx c
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General Form of Higher order LinearDifferential Equations
General form of second order Linear Differentialequation is
where P and Q are functions of x or constants
General form of n th order Linear Differential equationis
--Where are constants or functions of x or
constants
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2
2
d y dy P Qy R
dx dx
1 2
0 1 2 11 2 .........n n n
n n n n n
d y d y d y dy a a a a y b
dx dx dx dx 0 1 1, ,.........., ,n a a a b
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Classification of Higher order LinearDifferential Equations
Higher orderLinear diff.Eqns
HomogenousLinear diff.Eqns
ConstantCoefficients
Variablecoefficients
Non-homogenousLinear diff.Eqns
ConstantCoefficients
VariableCoefficients
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General Form of Higher order Linear DifferentialEquations
General form of second order Linear Differentialequation is
where P and Q are functions of x or constants
General form of n th order Linear Differentialequation is
--Where are constants or functions of x
or constants
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2
2
d y dy P Qy R
dx dx
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1 2
0 1 2 11 2 .........
n n n
n n n n n
d y d y d y dy a a a a y b dx dx dx dx
0 1 1, ,.........., ,n a a a b
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Note---
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This is another form of nth order lineardifferential equation
-----------------(1)where are functions of x or constants
.Here in eqn-1 left hand side we have n+1coefficients but we will divide eqn-1 by andtherefore we get there n-coefficients so in lineardiff.eqn the no. of coefficients is equal to theorder of diff.eqn
1
0 1 11 .........n n
n n n n
d y d y dy a a a a y b
dx dx dx
0 1 1, ,.........., ,,n n a a a a b
0a
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Homogenous and Non-homogenousLinear differential Eqn. of higher Order
----------(1)
If b = 0 in Eqn-1 then it is called Homogenouslinear differential eqn. And b is non-zero inEqn-1 that means either b is a function of x orconstant then it is called non-homogenouslinear differential eqn.
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1 2
0 1 11 2 .........n n n
n n n n
d y d y d y dy a a a b
dx dx dx dx
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Homogenous and Non-homogenous Linear differential Eqn. ofhigher Order with constant and variable Coefficients
if are functions of x in the followingequation
then that equation is called Linear differential eqn.with variable coefficients.
If are constants in given eqn.then it is
called Linear differential eqn. of higher Order withconstant coefficients.
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1 2
0 1 11 2 .........n n n
n n n n
d y d y d y dy a a a b
dx dx dx dx
0 1 1, ,.........., ,n a a a b
0 1 1, ,.........., ,n a a a b
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Linear combination of functions
Let are functions and
are constants then the expression
Is called linear combination of functions
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1 1 2 2( ) ( ) ...... ( )n n f x c f x c f x c
1 2, , ......., n
c c c
1 2( ), ( ), .... ( )n f x f x f x
1 2( ), ( ), .... ( )n f x f x f x
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Linear independent and Dependentfunctions
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Let are functions andare constants and If
and all
then the given functions are called linearlyindependent functions and if
and at least one then the givenfunctions are called linearly dependent functions
1 2( ), ( ), .... ( )n f x f x f x
1 2, , ......., n
c c c
1 1 2 2 0( ) ( ) ...... ( )n n f x c f x c f x c
1 2 0
, , ......., n c c c
1 1 2 2 0( ) ( ) ...... ( )n n f x c f x c f x c 0
i c
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Wronskian Test If then given functions
are Linearly Independent and
if are solutions of given differentialequation and
then given functions are linearly dependent but for functions whichare not solutions of given Diff.Eqn and
then they may or may not be linearly dependent
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1 2 0, , ........,( )n W f f f
1 2( ), ( ), .... ( )n f x f x f x
1 2( ), ( ), .... ( )n f x f x f x
1 2 0, ,........,( )n W f f f
1 2 0, ,........,( )n W f f f
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Example
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Using Wronskian test check whether givenfunctions f(x)=sinx and g(x) =cosx are linearlyindependent or dependent?
Using Wronskian test check whether givenfunctions
are linearly independent or dependent?
( ) , ( )x x f x e g x e
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Solution of homogenous lineardifferential equations of higher order
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Superposition Principle for Linearity
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Find Second Solution if one isgiven(Method of reduction of order)
If is the one solution of second orderhomogenous linear differential equation
Then second sol. where
this method is also known as method ofreduction of order
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1 y
0 y py qy
2 1 y uy
2
1
1 pdxu e dx
y
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Solution of homogeneous linear differential
here
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2e and e .mx mx mx y y me y m2 20 e e e 0 0.mx mx mx y ay cy m am b m am b
To solve the equation y +ay +by =0 substitute y = emx and try to determine m so thatthis substitution is a solution to the differential equation.
Compute as follows:
Homogeneous linear second order differential equations can always be solved bycertain substitutions.
This follows since e mx 0 for all x .
The equation m 2 + am + b = 0 is the Characteristic Equation or Auxiliaryequation of the differential equation y + ay + by = 0.
A differential equation of the typey +ay +by =0, a ,b real numbers,
is a homogeneous linear second order differentialequation with constant coefficients .
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So here we get two roots of Auxiliary eqn. and comparing them with
we get a==1,b= a and c=b and therefore
therefore roots are
Now if then CASE---1 CE m 2+am +b=0 has two different realsolutions m 1 and m 2
1 2
1 2e em x m x y C C
In this case the functions y = e m 1 x and y = e m 2 x are both solutions to the originalgiven differential equation and the general solution is
Example 0y y CE2 1 0m 1 or m 1.m
1 2e e x x y C C General Solution
The fact that all these functions are solutions can be verified by a direct calculation.
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2 0ax bx c2 24 4b ac a b
2 2
1, 2
4 42 2
b b ac a a bm m
a
0
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Solving Homogeneous 2 nd Order LinearEquations: Case II
CE has real double root m(that means equalroot) is one solution of equation now
Then second sol. And
In this case the functions y = emx and y =
x e mx are both solutions to the originalequation and the general sol.is
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mx y e
2 1 y uy
21
1 1 pdx axaxu e dx e dx x y e
1 2e e x x y C C x
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Solving Homogeneous 2 nd Order LinearEquations: Case III
Now,auxi.eqn has two complex solutions
Now, by Eulers formula So, our solutions
So , the general sol. is
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1 2,m m p iq
cos sini
e i
1
2
( )
( )
(cos sin )
(cos sin )
m x p iq x px iqx px
m x p iq x px iqx px
e e e e e qx i qx
e e e e e qx i qx
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1 21 1
m x m x y c e c e
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So, General solution in case-3 is
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1 2
1 2 1 2
1 2 1 2
1 2
(cos sin ) (cos sin )
[( cos cos ) ( sin sin )]
[( )cos ( ) sin )][ cos sin ]
px px
px
px
px
y c e qx i qx c e qx i qx
e c qx c qx c i qx c i qx
e c c qx c c i qx
e c qx c qx
1 2[ cos sin ] px y e c qx c qx
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Solution of non- homogeneous lineardiff.eqn.with constant coefficients
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The General Solution of non-
homogeneous linear diff.eqn.withconstant coefficients is of the form
Y = Complimentary function +Particular integral
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The meaning of particular integral
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General method Shortcut methods Method of variation Of parameters Method of undetermined coefficients
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Methods of finding Particular integral