Use the annihilator method to solve 2=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Title: Solving Differential Equations Using the Annihilator Method**
**Problem Statement:**
Use the annihilator method to solve the following differential equation with initial conditions:
\[ y'' + 4y = 2x + 3\cos(2x) \]
**Initial Conditions:**
\[ y(0) = 2, \]
\[ y'(0) = 0 \]
**Procedure:**
1. **Determine the Annihilator for the Non-Homogeneous Term:**
The non-homogeneous term in the differential equation is \( 2x + 3\cos(2x) \). We need to find the annihilators for each part of this term separately:
- For \( 2x \): The annihilator is \( D^2 \) where \( D = \frac{d}{dx} \).
- For \( 3\cos(2x) \): The annihilator is \( (D^2 + 4) \) because \( \cos(kx) \) is annihilated by \( D^2 + k^2 \) and here \( k=2 \).
2. **Combine the Annihilators:**
The combined annihilator for the non-homogeneous term \( 2x + 3\cos(2x) \) is the product of the individual annihilators:
\[ (D^2)(D^2 + 4) \]
3. **Apply the Annihilator to the Differential Equation:**
Applying the combined annihilator to both sides of the given differential equation results in:
\[ (D^2)(D^2 + 4)y = (D^2)(D^2 + 4)(2x + 3\cos(2x)) \]
Simplifying the right side (which should become zero):
\[ (D^2 + 4)(D^2)y = 0 \]
4. **Solve the Homogeneous Equation:**
Solve the homogeneous differential equation:
\[ (D^2 + 4)(D^2)y = 0 \]
Solving this, we find the complementary function (CF):
\[ y_c(x) = C_1 + C_2x + C_3 \cos(2x) + C_4 \sin(2x) \]
5. **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72bb43eb-cb6a-499b-b213-8a074a437ef9%2F2095e4bc-8a1f-46ab-98be-37e8403174fd%2Fvu1ghgp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Differential Equations Using the Annihilator Method**
**Problem Statement:**
Use the annihilator method to solve the following differential equation with initial conditions:
\[ y'' + 4y = 2x + 3\cos(2x) \]
**Initial Conditions:**
\[ y(0) = 2, \]
\[ y'(0) = 0 \]
**Procedure:**
1. **Determine the Annihilator for the Non-Homogeneous Term:**
The non-homogeneous term in the differential equation is \( 2x + 3\cos(2x) \). We need to find the annihilators for each part of this term separately:
- For \( 2x \): The annihilator is \( D^2 \) where \( D = \frac{d}{dx} \).
- For \( 3\cos(2x) \): The annihilator is \( (D^2 + 4) \) because \( \cos(kx) \) is annihilated by \( D^2 + k^2 \) and here \( k=2 \).
2. **Combine the Annihilators:**
The combined annihilator for the non-homogeneous term \( 2x + 3\cos(2x) \) is the product of the individual annihilators:
\[ (D^2)(D^2 + 4) \]
3. **Apply the Annihilator to the Differential Equation:**
Applying the combined annihilator to both sides of the given differential equation results in:
\[ (D^2)(D^2 + 4)y = (D^2)(D^2 + 4)(2x + 3\cos(2x)) \]
Simplifying the right side (which should become zero):
\[ (D^2 + 4)(D^2)y = 0 \]
4. **Solve the Homogeneous Equation:**
Solve the homogeneous differential equation:
\[ (D^2 + 4)(D^2)y = 0 \]
Solving this, we find the complementary function (CF):
\[ y_c(x) = C_1 + C_2x + C_3 \cos(2x) + C_4 \sin(2x) \]
5. **
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