Final Lecture 6 Maths-3

8/12/2019 Final Lecture 6 Maths-3

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Higher Order Linear Differential

Equations

Prepared 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 unknown

function and its derivatives. Examples are

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36

4

3

3

y

dx

dy

dx

yd

0z z

x y


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Differential equations can be

classified in two parts

(1) Ordinary differential equations

(2) Partial differential equations

An Ordinary differential equation is anequation which involves ordinary derivatives

is an example of O.D.E

A partial differential equation is an equationwhich involves partial derivatives

Is an example of P.D.E

36

4

3

3

y

dx

dy

dx

yd

0z z

x y


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Order of Differential Equation

The order of the differential equation is the

order of the highest order derivative in the

differential equation

Differential Equation ORDER

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32 xdx

dy

1

0932

2

ydx

dy

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 order

derivative term in the differential equation.

Differential Equation Degree

032

2

aydx

dy

dx

yd

03

53

2

2

dxdy

dxyd

1

3


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Solution of a differential equation

Definition--The solution of a differential

equation is a function which satisfies given

equation

types of solutions of differential equations

1) General solution

2) Particular Solution

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General Solution- The solution of differential equation

is called general if the no. of arbitrary constants equals

to the order of differential equationParticular solution- If we assign particular value to

arbitrary constant in general solution then it is called

particular 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=3x+c then its a particularsolution.


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Linear differential equations

A differential equation is called linear if

unknown function and the derivative of

unknown function are of first degree and they

are not multiplied together

A Differential equation of the form

is called linear differential equation of the first

order where either P and Q are functions of x

or constant

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dyPy Q

dx


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Solution of first order Linear

differential equations

The solution of first order linear diff.eqn. can

be 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 Linear

Differential Equations

General form of second order Linear Differentialequation is

where P and Q are functions of x or constants

General form of nthorder 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 nn n n

d y d y d y dy a a a a y b

dx dx dx dx

0 1 1, ,.........., ,

na a a b


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Classification of Higher order Linear

Differential 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 Differential

Equations

General form of second order Linear Differentialequation is

where P and Q are functions of x or constants

General form of nthorder Linear Differentialequation is

--

Where are constants or functions of xor 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 nn n n

d y d y d y dy a a a a y b

dx dx dx dx

0 1 1, ,.........., ,

na 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 nn n

d y d y dy a a a a y b

dx dx dx

0 1 1, ,.........., ,,

n na a a a b

0a


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Homogenous and Non-homogenous

Linear differential Eqn. of higher Order

----------(1)

If b = 0in Eqn-1 then it is called Homogenouslinear differential eqn. And b is non-zero in

Eqn-1 that means either b is a function of x or

constant then it is called non-homogenous

linear differential eqn.

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1 2

0 1 11 2 .........

n n n

nn 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. of

higher Order with constant and variable Coefficients

if are functions of x in the following

equation

then that equation is calledLinear 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

nn n n

d y d y d y dy a a a b

dx dx dx dx

0 1 1, ,.........., ,

na a a b

0 1 1, ,.........., ,

na 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 nf x c f x c f x c

1 2, , ......., nc c c

1 2( ), ( ), .... ( )

nf x f x f x

1 2( ), ( ), .... ( )

nf x f x f x


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Linear independent and Dependent

functions

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Let are functions and

are 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( ), ( ), .... ( )

nf x f x f x

1 2, , ......., nc c c

1 1 2 2 0( ) ( ) ...... ( )

n nf x c f x c f x c

1 2 0

, , ......., nc c c

1 1 2 2 0( ) ( ) ...... ( )n nf x c f x c f x c 0

ic



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Wronskian Test for Linearly

Independent -Dependent

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Let are n functions and

their first n-1 derivatives exists then the

Wronskian or Wronski Determinant is denoted

by and defined as

1 2( ), ( ), .... ( )

nf x f x f x

1 2, , ........,( )

nW f f f

1

1

1

1 1

1

' '

'' ''

( ) ( )

....

....

....

....................

....................

.......

n

n

n

n n

n

f f

f f

f f

f f


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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, , ........,( )nW f f f

1 2( ), ( ), .... ( )

nf x f x f x

1 2( ), ( ), .... ( )

nf x f x f x

1 2 0, ,........,( )nW f f f

1 2 0, ,........,( )nW f f f


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Example

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Using Wronskian test check whether given

functions f(x)=sinx and g(x) =cosx are linearly

independent or dependent?

Using Wronskian test check whether given

functions

are linearly independent or dependent?

( ) , ( )x x

f x e g x e


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Solution of homogenous linear

differential equations of higher order

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Superposition Principle for Linearity


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Find Second Solution if one is

given(Method of reduction of order)

If is the one solution of second order

homogenous linear differential equation

Then second sol. where

this method is also known as method of

reduction of order

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1y

0y py qy

2 1y 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 m

2 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= emxand 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 by

certain substitutions.

This follows since emx0 for allx.

The equation m2+ am+ b= 0 is the Characteristic Equation or Auxiliary

equation of the differential equation y + ay + by= 0.

23

A differential equation of the type

y+ay+by=0, a,breal numbers,is a homogeneous linear second order differential

equation 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 m2+am+b=0 has two different real

solutions m1 and m2

1 21 2e e

m x m x y C C

In this case the functions y = em1x and y = em2x are both solutions to the original

given differential equation and the general solution is

Example 0y y CE 2

1 0m 1 or m 1.m

1 2e e

x xy 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 c 2 24 4b ac a b

2 2

1, 2

4 4

2 2

b b ac a a bm m

a

0

Sci.& Hum.Dept. ,E.M.-3 24


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Solving Homogeneous 2ndOrder Linear

Equations: Case II

CE has real double root m(that means equal

root) is one solution of equation now

Then second sol. And

In this case the functions y = emx and y =

xemx are both solutions to the originalequation and the general sol.is

25

mxy e

2 1y uy

2

1

1 1pdx axax

u e dx e dx xy e

1 2

e ex xy C C x

3/4/2014Sci.& Hum.Dept. ,E.M.-319/07/2013


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Solving Homogeneous 2ndOrder Linear

Equations: 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

Sci.& Hum.Dept. ,E.M.-319/07/2013

1 2

1 1

m x m xy 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 ]pxy e c qx c qx



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Solution of non- homogeneous linear

diff.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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General method

Shortcut methods

Method of variation Of parameters

Method of undetermined coefficients

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Methods of finding Particular integral

Final Lecture 6 Maths-3

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