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
3.6 Cauchy-Euler Equations
Learning Outcome
At the end of this section, you should be ableto:
Identify and solve a Cauchy-Eulerdifferential equation
A linear Cauchy-Euler differential equation
���� ���
��� + �������� �����
����� +⋯+ ��� ���� + �� = �(�)
The coefficients �� , ���� , … ,� are constants
The degree of � is the same as the order of the derivatives
same same Function in terms of �
3.6 Cauchy-Euler Equations
same
A linear second order Cauchy-Euler differential equation
same same Function in terms of �
3.6 Cauchy-Euler Equations
��� ���
��� + �� ���� + �� = �(�)
Homogeneous Cauchy-Euler differential equation
3.6 Cauchy-Euler Equations
��� ���
��� + �� ���� + �� = 0
The general solution is
� = ��
3.6 Cauchy-Euler Equations
��� ���
��� + �� ���� + �� = �(�)Non-Homogeneous Cauchy-Euler differential equation
The general solution is
� = �� + ��
�� is obtained by solving the associated homogeneous
�� is obtained by using variation of parameters
��� ���
��� + �� ���� + �� = 0
��� ���
��� + �� ���� + �� = �(�)
3.6 Cauchy-Euler Equations
������ + ���� + �� = 0
�� = �����
��� = � � − 1 ����
� = ��
3.6 Cauchy-Euler Equations
Method of Solution
Step 1: Let the solution of the DE be
���� � − 1 ���� + ������� + ��� = 0
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
������ + ���� + �� = 0
�� � − 1 �� + ���� + ��� = 0��(� − 1) + �� + � �� = 0
��� + � − � � + � = 0
Auxiliary Equation
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
������ + ���� + �� = 0
��� + � − � � + � = 0
DE
AE
Step 3: Solve the AE
Step 4: Find the general solution
The general solution is
Case 1: Distinct and Real Roots
�� ≠ �
3.6 Cauchy-Euler Equations
� = ��!�� + � !�
The general solution is
Case 2: Repeated Real Roots
�� = �
3.6 Cauchy-Euler Equations
� = ��!�� + � !�� "# !
The general solution is
Case 3: Conjugate Complex Roots
�� = $ + %& , � = $ − %&
3.6 Cauchy-Euler Equations
� = !$ �� '() & "# ! + � )*# & "#!
Solve �� +,-+., − 2� +-+. − 4� = 0
� = ���� = �����
��� = � � − 1 ����
Example 1
3.6 Cauchy-Euler Equations
Solution
Step 1: Let the solution of the DE be
�� − 3� − 4 �� = 0
��� �− 1 ���� − 2������ − 4�� = 0� � − 1 �� − 2��� − 4�� = 0
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
�� ���
��� − 2� ���� − 4� = 0
�� − 3� − 4 = 0��� + � − � � + � = 0
�� = −1 , �� = 4
= ����� + ���2
3.6 Cauchy-Euler Equations
Step 3: Solve the AE
�� − 3� − 4 = 0Case 1
Step 4: Find the general solution
The general solution is � = ����3 + ����,
� = ���� = �����
��� = � � − 1 ����
Example 2
3.6 Cauchy-Euler Equations
Solve 4�� +,-+., + 8� +-+. + � = 0
Solution
Step 1: Let the solution of the DE be
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
��� + � − � � + � = 0
4�� ���
��� + 8� ���� + � = 0
4�� + 4� + 1 = 0
4��� �− 1 ���� + 8������ + �� = 04� � − 1 �� + 8��� + �� = 0
= ������ + ���
��� ln�
3.6 Cauchy-Euler Equations
Step 3: Solve the AE
Case 2
Step 4: Find the general solution
The general solution is � = ����3 + ����3 ln �
4�� + 4� + 1 = 02� + 1 � = 0
�� = �� = −12
� = ���� = �����
��� = � � − 1 ����
Example 3
3.6 Cauchy-Euler Equations
Solve 4�� +,-+., + 17� = 0Solution
Step 1: Let the solution of the DE be
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
��� + � − � � + � = 0
4�� − 4� + 17 = 0
4��� �− 1 ���� + 17�� = 04� � − 1 �� + 17�� = 0
4�� ���
��� + 17� = 0
= ��� �� cos 2 ln � + �� sin 2 ln �
3.6 Cauchy-Euler Equations
Step 3: Solve the AE
Case 3
Step 4: Find the general solution
The general solution is � = �< �� cos = ln � + �� sin = ln �
�� = 12 + 2>, �� = 1
2 − 2>4�� − 4� + 17 = 0
� = ���� = �����
��� = � � − 1 ����
Example 4
3.6 Cauchy-Euler Equations
Solve �@ +A-+.A + 5�� +,-+., + 7� +-+. + 8� = 0
���� = � � − 1 (� − 2)���@
Solution
Step 1: Let the solution of the DE be
3.6 Cauchy-Euler Equations
Step 2: Substitute into DE
�@ �@�
��@ + 5�� ���
��� + 7� ���� + 8� = 0
�@� �− 1 � − 2 ���@ + 5��� �− 1 ����+7������ + 8�� = 0
�� � � − 1 � − 2 + 5� � − 1 + 7� + 8 = 0�� �@ + 2�� + 4� + 8 = 0
�@ + 2�� + 4� + 8 = 0
3.6 Cauchy-Euler Equations
Step 3: Solve the AE
Case 3
Step 4: Find the general solution
The general solution is
�@ + 2�� + 4� + 8 = 0(� + 2)(�� + 4) = 0
�� = −2 , �� = 2> , �@ = −2>Case 1
� = ����3 + �< �� cos = ln � + �@ sin = ln �= ����� + �� cos 2 ln � + �@ sin 2 ln �
Example 5
3.6 Cauchy-Euler Equations
Solve ����� − 3��� + 3� = 2�2D.SolutionStep 1: Find the complementary solution
Solve the associated homogeneous equation
Change to auxiliary equation.
����� − 3��� + 3� = 0
�� − 4� + 3 = 0
= ��� + ���@
3.6 Cauchy-Euler Equations
The roots of the auxiliary equation are
�� − 4� + 3 = 0
� − 1 � − 3 = 0�� = 1 , �� = 3 Case 1
The complementary solution is
�� = ����3 + ����,
3.6 Cauchy-Euler Equations
Step 2: Find the particular solution
Step 2.1: Identify �� and ���� = ��� + ���@�� = � , �� = �@
Step 2.2: DE in standard form and identify E ������ − 3��� + 3� = 2�2D.
��� − 3� �� +
3�� � = 2��D.
E(�) = 2��D.
3.6 Cauchy-Euler Equations
F�� = G�G = −��E �
G F�� = G�G = ��E �
G
Step 2.3: Compute
and
G� =? G� =?G =?
G(�, �@) = � �@1 3��
= 2�@
3.6 Cauchy-Euler Equations
G� = 0 �@2��D. 3��
= −2�ID.G� = � 0
1 2��D.= 2�@D.
G� = 0 ��E(�) ��� G� = �� 0
��� E(�)
F�� = −2�ID.2�@
= −��D.
F�� = 2�@D.2�@
= D.
3.6 Cauchy-Euler Equations
F�� = G�G F�� = G�
G
Step 2.4: Integrate
F�� = −��D. F�� = D.
F� = −J��D. ��
= −��D. + 2�D. − 2D.F� = JD. ��
= D.
3.6 Cauchy-Euler Equations
Step 2.5: Particular solution
�� = F� � �� � + F�(�)��(�)
Step 3: General solution
The general solution is
� = �� + ��
= −��D. + 2�D. − 2D. � + D.�@= 2��D. − 2�D.
= ��� + ���@ + 2��D. − 2�D.
3.6 Cauchy-Euler Equations
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