Ordinary Differential Equations[FDM 1023]

Linear Higher-Order Differential Equations

Chapter 3

Overview

Chapter 3: Linear Higher-Order Differential Equations

3.1. Definitions and Theorems

3.2. Reduction of Order

3.3. Homogeneous Linear Equations with

Constant Coefficients

3.4. Undetermined Coefficients

3.5. Variation of Parameters

3.6. Cauchy-Euler Equations

3.1. Definitions and Theorems

Learning Outcomes

At the end of this section you should be able to know:

1)The basic theorems on existence and uniqueness

of solutions of DE (based on IVP and BVP)

2) The definitions of linear dependence, linear

independence, Wronskian, fundamental set of

solution, homogeneous DE and non-homogeneous

DE.

An n-th order IVP for a linear differential equations

will have the following form

+

++

+ = ()

subject to = , = ,

=

3.1. Definitions and Theorems

Theorem

Existence and Uniqueness of Solution for IVP

3.1. Definitions and Theorems

If there is an interval containing on which:

1) are continuous

2)

I

, , , , and()

0

Then, a solution of the IVP EXISTS on and is

UNIQUE.

I

Show that

2 + 9 = , 1 = 0, 1 = 0

has the unique solution = 0 on the interval (,)

Solution:

The coefficients of , and are the functions 2, 9, and = which are all continuous function on , .

The coefficients of the highest derivative is 2 which is a non-zero function.

All conditions of existence of a unique solution are satisfied.

By theorem, = 0 is the unique solution on , .

3.1. Definitions and Theorems

Example 1

Solution:

The IVP 3 + 5 + 7 = 0, 1 = 0, 1 = 0,

1 = 0 possesses the unique solution = 0. Why?

All the coefficients of , , and are non zero.

The coefficients of the highest derivative is 3 which is a non-zero function.

All conditions of existence of a unique solution are satisfied.

Hence, = 0 is the only solution on any interval containing = 1.

3.1. Definitions and Theorems

Example 2

A BVP can have zero, one or infinitely manysolutions.

+

+ = ()

= , =

A problem such as

subject to

where the dependent variable # or its derivatives arespecified at different points is called BOUNDARYVALUE PROBLEM (BVP).

3.1. Definitions and Theorems

Definition

Solution for BVP

Solution:

Consider + 16 = 0 with its general solution = % cos 4 + % sin 4 on ,

We will look at several boundary conditions for this DE

1) 0 = 0, +

= 1

0 = % cos 4(0) + % sin 4 0 = 0 % = 0

,

2= % cos 4(

,

2) + % sin 4

,

2= 1 % = 1

There are two contradictory values for %

Conclusion : There is no solution (zero solution)

3.1. Definitions and Theorems

Example

2) 0 = 0, +

-= 0

0 = % cos 4(0) + % sin 4 0 = 0 % = 0

,

8= % cos 4(

,

8) + % sin 4

,

8= 0 % = 0

The solution of the BVP is = 0

Conclusion : There is only one solution

3.1. Definitions and Theorems

Consider + 16 = 0 with its general solution = % cos 4 + % sin 4 on ,

3) 0 = 0, +

= 0

0 = % cos 4(0) + % sin 4 0 = 0 % = 0

,

2= % cos 4(

,

2) + % sin 4

,

2= 0 % = 0

The solution of the BVP is = % sin 4 where % is any

number ( is a 1-parameter family of solutions)

Conclusion : There is infinitely many solutions

3.1. Definitions and Theorems

Consider + 16 = 0 with its general solution = % cos 4 + % sin 4 on ,

A set of functions / , / , , /() is said to be

linearly dependent on an interval , if there existconstants %, % , , % (not all zero) such that

for every in the interval, .

%/ + %/ ++ %/ = 0

If the set of functions is not linearly dependent on the interval, it is said to be linearly independent.

3.1. Definitions and Theorems

Definition

I

I

Linearly Dependent / Independent

Suppose each of the functions / , / , , /()

possesses at least 0 1 derivatives.

The determinant

is called the Wronskian of the functions

3.1. Definitions and Theorems

Definition

( )

( ) ( ) ( )112

1

1

21

21

21,...,,

=

n

n

nn

n

n

n

fff

fff

fff

fffW

L

MMMM

L

L

Wronskian of the Functions

1) If 1 /, /, , / = 0 then the set of functions

/ , / , , /() is LINEARLY DEPENDENT.

2) If 1 /, /, , / 0 then the set of functions

/ , / , , /() is LINEARLY INDEPENDENT.

3.1. Definitions and Theorems

Theorem

Solution: Use the Wronskian

Determine whether the set of functions = cos and = sin is linearly dependent or not.

W , =

=cos sin

sin cos

= cos sin

= 1

The set of functions is linearly independent.

3.1. Definitions and Theorems

Example 1

0

Show that the set of solution = , = and

3 = 4 3 is linearly dependent .

W , , 3 =

3

3

3

= 4 3

1 2 4 6

0 2 6

3.1. Definitions and Theorems

Example 2

Solution: Use the Wronskian

= 4 3

1 2 4 6

0 2 6

= 2 4 6

2 61

4 3

2 6

= 12 8 + 12 6 8 + 6

= 0

The set of functions is linearly dependent.

3.1. Definitions and Theorems

+0 4 3

2 4 6

A linear 0-th order DE of the form

456

45+

45786

4578++

46

4+ = 0

is said to be homogeneous.

is a homogeneous linear second-order DE.

3.1. Definitions and Theorems

Definition

Example

0532 =+ yyy

Homogeneous DE

A linear 0-th order DE of the form

456

45+

45786

4578++

46

4+ = ()

is said to be non-homogeneous.

3.1. Definitions and Theorems

Definition

Example

2532 xyyy =+

is a non-homogeneous linear second-order DE.

Non-Homogeneous DE

A set of , , , of a linearly independentsolutions of an 0 -th order homogeneous linear

equation on an interval is said to be a set offundamental set of solutions on the interval.

3.1. Definitions and Theorems

Definition

I

Fundamental Set of Solutions

Let , , , be a fundamental set of solutions of an 0-th order homogeneous linear DE on an interval .

For any 0 th order homogeneous linear DE, there is a fundamental set of solutions on an interval .

The general solution of the DE in an interval is

= % + % ++ %

where are arbitrary constant.

3.1. Definitions and Theorems

Theorem

I

I

I

nici ,..,1, =

General Solution Homogeneous Linear DE

Let be a fundamental set of solutions the associated homogeneous linear DE on an interval .

The general solution of the DE on the interval is

where are arbitrary constants.

nyyy ,...,, 21I

nici ,..,1, =

I

3.1. Definitions and Theorems

Definition

= % + % ++ % +9

General Solution Non-Homogeneous Linear DE

Any solution free of arbitrary parameters, that satisfies the DE is said to be a particular solution.

py

3.1. Definitions and Theorems

Homogeneous DE

= % + % ++ % +9

= % + % ++ %

Non-Homogeneous DE

Determine whether the given set of functions is linearly dependent or linearly independent on the interval ,

1. = 1, = , 3 =

2./ = 1 + , / = , /3 =

3.1. Definitions and Theorems

Exercise 3.1