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Advanced Engineering Mathematics
Comprehensive Class Notes
Chapter 4: Nonhomogeneous Linear Differential Equations
Know-How | Student Edition
, 2
Chapter Overview
In Chapter 4, we extend ideas from Chapter 3. Now the equation contains an external
forcing term, representing input, load, or excitation from the environment.
Standard form:
a y″ + b y′ + c y = F(x)
Examples of forcing terms F(x):
• • polynomial
• • exponential
• • sine or cosine
Goal: find the general solution
y = y_h + y_p
where y_h is complementary (from Chapter 3) and y_p is any particular solution.
1. Method of Undetermined Coefficients
Useful when F(x) is polynomial, exponential, or sinusoidal.
Idea: guess form of solution and substitute.
Example: y″ + y = 5
Complementary: y_h = C₁ cos x + C₂ sin x
Guess constant particular: y_p = A → substitute → A = 5
Final: y = C₁ cos x + C₂ sin x + 5
2. Variation of Parameters (Idea Only)
This powerful method works when F(x) is more complicated.
Instead of constants C₁ and C₂, we allow them to vary with x.
Know-How | Student Edition
Advanced Engineering Mathematics
Comprehensive Class Notes
Chapter 4: Nonhomogeneous Linear Differential Equations
Know-How | Student Edition
, 2
Chapter Overview
In Chapter 4, we extend ideas from Chapter 3. Now the equation contains an external
forcing term, representing input, load, or excitation from the environment.
Standard form:
a y″ + b y′ + c y = F(x)
Examples of forcing terms F(x):
• • polynomial
• • exponential
• • sine or cosine
Goal: find the general solution
y = y_h + y_p
where y_h is complementary (from Chapter 3) and y_p is any particular solution.
1. Method of Undetermined Coefficients
Useful when F(x) is polynomial, exponential, or sinusoidal.
Idea: guess form of solution and substitute.
Example: y″ + y = 5
Complementary: y_h = C₁ cos x + C₂ sin x
Guess constant particular: y_p = A → substitute → A = 5
Final: y = C₁ cos x + C₂ sin x + 5
2. Variation of Parameters (Idea Only)
This powerful method works when F(x) is more complicated.
Instead of constants C₁ and C₂, we allow them to vary with x.
Know-How | Student Edition
