differential equations mth 242 lecture # 09 dr. manshoor ahmed

Differential EquationsMTH 242

Lecture # 09

Dr. Manshoor Ahmed

Summary (Recall)

• Higher order linear differential equation.

• Homogeneous and non-homogeneous equations with constant coefficients.

• Initial value problem (IVP) and it’s solution.

• Existence and Uniqueness of Solutions.

• Boundary value problem (BVP) and it’s solution.

• Linear independence and dependence of functions.

• Wronskian of a set of functions.

Solution of Higher Order Linear Equation


Preliminary Theory

• A linear nth order equation of the form

is said to be homogeneous.• A linear nth order equation of the form

,

where g(x) is not identically zero, is said to be non-homogeneous.

)()()()()( 011

11 xgyxa

dxdyxa

dx

ydxadx

ydxan

nnn

nn

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n


• For solution of non-homogeneous DE, we first solve the associated homogeneous differential equation

.

• So, first we learn how to solve the homogeneous DE.• As function y=f(x) that satisfies the associated homogeneous equation

is called solution of the differential equation.

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n


Theorem: (Superposition principle Homogeneous equations)

Let y1,y2,…,yk be solutions of homogeneous linear nth order differential equation on an interval I then the linear combination ,

where the ci, i = 1,2,…,k are arbitrary constants, is also a solution on the interval.

Remarks:i. A constant multiple y=c1 y1(x) of a solution y1(x) of a homogeneous DE is also a solution.ii. A homogeneous linear differential equation always possesses the trivial solution y = 0.

)(......)()( 2211 xycxycxycy kk


The superposition principle is valid of linear DE and it does not hold in case of non-linear DE.

Example 1:The functions y1(x) = x2 and y1(x) = x2 lnx are both solutions of

homogeneous equation

on

Then by Superposition principle

is also the solution of the DE on the interval.

042 ''''2 yxyyx

xxcxcy ln22

21

),0(


Example 2The functions

all satisfy the homogeneous DE

on .

Thus are all solutions of the given DE.Now suppose that

xxx eyeyey 33

221 ,,

06116 2

2

3

3

ydxdy

dxyd

dxyd

xxx ecececy 33

221

),(


Then,

Therefore,

Thus, is a solution the DE.

xxx

xxx

xxx

ecececdx

yd

ecececdx

yd

ecececdxdy

33

2213

3

33

2212

2

33

221

278

94

32

ydxdy

dxyd

dxyd 6116 2

2

3

3

0)6060()3030()1212(

)6335427()622248()6116(3

32

21

33333

222221

xxx

xxxxxxxxxxxx

ececec

eeeeceeeeceeeec

xxx ecececy 33

221


Linearly independent solutionsTheorem:

Let y1,y2,…, yn be solutions of homogeneous linear nth order DE

on an interval I . Then the set of solutions is linearly independent on I if and only if

for every x in the interval.

0),...,,( 21 nyyyW

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n


In other wordsThe solutions y1,y2,…, yn are linearly dependent if and only if

For example, we consider a second order homogeneous linear DE

and Suppose that y1,y2 are two solutions on an interval I.

Then either

Or

IxyyyW n ,0),...,,( 21

0012

2

2 yadxdya

dxyda

IxyyW ,0),( 21

IxyyW ,0),( 21


To verify this we write the equation as

Now

Differentiating w.r.t x , we have

Since y1 and y2 are solutions of the DE

02

2

QydxdyP

dxyd

2'1

'21'

2'1

2121 ),( yyyy

yyyy

yyW

2''

1''

21 yyyydx

dW

02

2

QydxdyP

dxyd


Therefore,

Multiplying 1st equation by y2 and 2nd by y1 then we have

Subtracting the two equations we have:

Or

01'1

''1 QyPyy

02'2

''2 QyPyy

0212'12

''1 yQyyPyyy

021'21

''21 yQyyPyyy

0)()( ''21

'21

''12

''21 yyyyPyyyy

0 PWdx

dW


This is a linear 1st order differential equation in W, whose solution is

Therefore

If then the solutions are linearly independent on I.If then the solutions are linearly dependent on I.

Pdx

ceW

0c

0c

IxyyW ,0),( 21

IxyyW ,0),( 21


Fundamental set of solutionsAny set y1,y2,…, yn of n linearly independent solutions of the homogeneous linear nth order DE (3) on an interval I is said to be a Fundamental set of solutions on the interval.

General Solution of homogeneous equations:Let y1,y2,…, yn be fundamental set of solutions of the homogeneous linear nth order differential equation (3) on an interval I, then the general solution of the equation on the interval is defined to be

where the ci, i = 1,2,…,n are arbitrary constants.

)(......)()( 2211 xycxycxycy nn

Solution of Higher Order Linear EquationExample 1

The functions y1(x) = e3x ,y2(x) = e2x and y3(x) = e3x satisfy DE

Since,

For x, y1, y2 , y3 every real value form a Fundamental set of solutions on and

is the general solution on the given interval.

06116 2

2

3

3

ydxdy

dxyd

dxyd

0293

42),,( 6

3

3

3

2

2

2

32 x

x

x

x

x

x

x

x

x

x

xxx eeee

ee

e

eee

eeeW

) ,( xxx ecececy 3

32

21


xexy 353sinh4

xexy 3' 153cosh12

)53sinh4(9

453sinh363''

3''

x

x

exy

exy

yy 9'' 09'' yyxexy 353sinh4

09'' yy


The general solution of the differential equation

is

If we take, .

We get

Hence, the particular solution has been obtained from the general solution.

09'' yyxx ececy 3

23

1

7.2 21 cc

x

xxx

xxx

xx

exy

eeey

eeey

eey

3

333

333

33

53sinh4

5)2

(4

522

72


Solution of Non-Homogeneous EquationsA function yp that satisfies the non-homogeneous differential equation

and is free of parameters is called the particular solution of the differential equation

Example 1

Suppose that

Then,

xxy p 3

xyxy pp 6,13 ''2'

)()()()()( 011

11 xgyxa

dxdyxa

dx

ydxadx

ydxan

nnn

nn


Therefore,

Hence,

is a particular solution of the differential equation

xxyxyyx

xxxxxxyxyyx

ppp

ppp

6482

)(8)13(2)6(823'''2

322'''2

xxy p 3

xxyxyyx ppp 6482 3'''2


Complementary functionThe general solution

of the homogeneous linear differential equation

is known as the complementary function for the non-homogeneous linear differential equation.

nnc ycycycy ...2211

)()()()()( 011

11 xgyxa

dxdyxa

dx

ydxadx

ydxan

nnn

nn

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n


General Solution Non-Homogeneous equationsSuppose thatThe particular solution of the non-homogeneous equation

is yp.• The complementary function of the non-homogeneous differential equation

is nnc ycycycy ...2211

)()()()()( 011

11 xgyxa

dxdyxa

dx

ydxadx

ydxan

nnn

nn

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n


•Then general solution of the non-homogeneous equation on the interval ‘I’is given by:

Or

Hence General Solution = Complementary Solution + Any Particular Solution

pc yyy

)()()()(...)()( 2211 xyxyxyxycxycxycy pcpnn


Example

Let is a particular solution of the equation

Then the complementary function of the associated homogeneous equation

is

xy p 21

1211

ydxdy

dxyd

dxyd 6116 2

2

3

3

xxx ecececy 33

221

xydxdy

dxyd

dxyd 36116 2

2

3

3


So the general solution is

Implies

)()( xyxyy pc

xecececy xxx

21

12113

32

21


Superposition Principle Non-Homogeneous Equations

Let be ‘k’ particular solutions of a linear nth order differential equation (4) on an interval I corresponding to ‘k’ distinct functions g1,g2,…, gk that is denotes a particular solution of the corresponding differential equation

where i = 1,2,…,k .Then

is a particular solution of

pkpp yyy ,...,, 21

)()()(...)()( 0'

1)1(

1)( xgyxayxayxayxa i

nn

nn

)(...)()( 21 xyxyxyy pkppp

piy

)(),...,(),()()(...)()( 210'

1)1(

1)( xgxgxgyxayxayxayxa k

nn

nn


ExampleConsider the differential equation

Suppose that

Then,

Therefore,

is a particular solution of the non-homogenous differential equation

xxeexxyyy xx 228241643 22'''

xp

xpp xexyexyxxy )(,)(,4)( 3

22

21

21

'1

''1 1624843 xxyyy ppp

21 4)( xxy p

8241643 2''' xxyyy


Similarly, it can be verified that

are particular solutions of the equations:

And

respectively. Hence,

is a particular solution of the differential equation

xp

xp xexyexy )( and )( 3

22

xeyyy 2''' 243

xxeyyy x 243 '''

xxpppp xeexxyxyxyy 22

321 4)()()(

xxeexxyyy xx 228241643 22'''

ExercisesVerify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.

ExercisesVerify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.

Summary

• Solution of Higher order linear differential equation.

• Solution of homogeneous equations.

• Superposition principle.

• Linearly independent Solutions.

• Fundamental set of solutions, general solution.

• Solution of non-homogeneous equations.

• Complementary function and particular solution.

differential equations mth 242 lecture # 09 dr. manshoor ahmed

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