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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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364

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

364

3

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

Sci.& Hum.Dept. ,E.M.-3

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

25

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

26

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