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

Learning Outcome

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

Solve the homogeneous linear ODE with

constant coefficients using the auxiliary

equation.

3.3 Homogeneous linear equations with constant coefficients

Consider the second order DE

���� + ��� + �� = 0

where �, �and� are constants.

3.3 Homogeneous linear equations with constant coefficients

Step 1: Let the solution of the DE be

� = ���

�� = ����

��� = �����

3.3 Homogeneous linear equations with constant coefficients

Method of Solution

Step 2: Substitute into DE

������ + ����� + ���� = 0

3.3 Homogeneous linear equations with constant coefficients

���� + ��� + �� = 0

��� ��� + �� + � = 0

��� + �� + � = 0 ��� ≠ 0

Auxiliary Equation

Step 2: Substitute into DE

������ + ����� + ���� = 0

3.3 Homogeneous linear equations with constant coefficients

���� + ��� + �� = 0

��� ��� + �� + � = 0

��� + �� + � = 0

DE

AE

Use the formula

� =−� ± �� − 4��

2�

�� − 4�� will lead into three different cases

3.3 Homogeneous linear equations with constant coefficients

Step 3: Solve the AE

1) Case 1 : �� − 4�� > 0

�� and �� distinct and real

2) Case 2 : �� − 4�� = 0

�� and �� repeated and real

3) Case 3 : �� − 4�� < 0

�� and �� conjugate complex

⇒ ��= ��

⇒ ��= � + �� , �� = � − ��

3.3 Homogeneous linear equations with constant coefficients

⇒ ��≠ ��

� = !�! + "�"

3.3 Homogeneous linear equations with constant coefficients

Step 4: Find the general solution

It has been assumed that the solution is � = ���

The general solution is

� = !#$!% + "#

$"%

3.3 Homogeneous linear equations with constant coefficients

Case 1: Distinct and Real Roots

$! ≠ $"

The general solution is

3.3 Homogeneous linear equations with constant coefficients

Case 2: Repeated Real Roots

$! = $"

� = !#$!% + "%#

$!%

The general solution is

3.3 Homogeneous linear equations with constant coefficients

Case 3: Conjugate Complex Roots

$! = & + '( , $" = & − '(

� = #&% ! )*+(% + " +,-(%

Solve ��� − 36� = 0

Solution

Step 1: Let the solution of the DE be

� = ���

�� = ����

��� = �����

3.3 Homogeneous linear equations with constant coefficients

Example 1

Step 2: Substitute into DE

��� − 36� = 0

��� �� − 36 = 0

�� − 36 = 0

� − 6 � + 6 = 0

�� = 6,�� = −6

= � �0� + � �10�

3.3 Homogeneous linear equations with constant coefficients

����� − 36��� = 0

Step 3: Solve the AE

�� − 36 = 0

Case 1

Step 4: Find the general solution

The general solution is

3.3 Homogeneous linear equations with constant coefficients

� = ����2� + ���

�3�

= ���0� + ���

10�

� = ���

�� = ����

��� = �����

3.3 Homogeneous linear equations with constant coefficients

Example 2

Solve ��� + 6�� + 9� = 0

Solution

Step 1: Let the solution of the DE be

3.3 Homogeneous linear equations with constant coefficients

Case 2

��� + 6�� + 9� = 0

����� + 6���� + 9��� = 0

��� �� + 6� + 9 = 0

�� + 6� + 9 = 0

�� + 6� + 9 = 0

� + 3 � + 3 = 0

�� = �� = −3

Step 2: Substitute into DE

Step 3: Solve the AE

The general solution is

3.3 Homogeneous linear equations with constant coefficients

� = ����2� + ��5�

�2�

= ���16� + ��5�

16�

Step 4: Find the general solution

� = ���

�� = ����

��� = �����

3.3 Homogeneous linear equations with constant coefficients

Example 3

Solve 2��� + 2�� + � = 0

Solution

Step 1: Let the solution of the DE be

3.3 Homogeneous linear equations with constant coefficients

2��� + 2�� + � = 0

2����� + 2���� + ��� = 0

��� 2�� + 2� + 1 = 0

2�� + 2� + 1 = 0

2�� + 2� + 1 = 0

� =−� ± �� − 4��

2�

Step 2: Substitute into DE

Step 3: Solve the AE

3.3 Homogeneous linear equations with constant coefficients

� =−2 ± 4 − 4(2)(1)

2(2)

Case 3

=−2 ± −4

4

=−2 ± 2�

4

=−1 ± �

2

�� =−1 + �

2

�� =−1

2+

�

2

�� =−1 − �

2

�� =−1

2−

�

2

Step 4: Find the general solution

The general solution is

3.3 Homogeneous linear equations with constant coefficients

Compare with �� = � + �� �� = � − ��

�� =−1

2+ �

1

2 �� =−1

2− �

1

2

� = −�

�, β =

�

�

� = �;� �� cos�5 + �� sin �5

= �1���

�� cos1

25 + �� sin

1

25

Higher-Order Equations

We can extend the three cases in the 2nd

order to higher order.

Only, need to know how to factorize theobtained auxiliary equation.

3.3 Homogeneous linear equations with constant coefficients

Solution

Step 1:

3.3 Homogeneous linear equations with constant coefficients

Example

Solve 043'''''

=−+ yyy

04323

=−+ mm

043'''''

=−+ yyy

Step 2:

3.3 Homogeneous linear equations with constant coefficients

Case 1

0)44)(1(2

=++− mmm

04323

=−+ mm

0)2)(1(2

=+− mm

Case 2

�� = 1 �� = �6 = −2

Step 3:

The general solution is

3.3 Homogeneous linear equations with constant coefficients

� = ����2� + ���

�3� + �65��3�

= ���� + ���

1�� + �65�1��