Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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. **
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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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