Solve the differential equation by variation of parameters. 1 9 + ex y(x) = y" + 3y' + 2y =
Solve the differential equation by variation of parameters. 1 9 + ex y(x) = y" + 3y' + 2y =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Solve the Differential Equation by Variation of Parameters**
Consider the differential equation:
\[ y'' + 3y' + 2y = \frac{1}{9 + e^x} \]
To solve this equation using the method of variation of parameters, follow these steps:
1. **Find the complementary solution (y_c):**
- Solve the homogeneous equation \( y'' + 3y' + 2y = 0 \).
2. **Particular solution (y_p) using variation of parameters:**
- Use the formula for variation of parameters to find a particular solution of the non-homogeneous equation.
3. **General Solution:**
- The general solution is \( y(x) = y_c(x) + y_p(x) \).
The solution will be displayed as:
\[ y(x) = \text{(general solution here)} \]
Complete the calculations to find the complete form of \( y(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76aaab3c-5a4a-4ff8-814d-d2e8a1f30217%2F8a56d331-5f30-4a3b-99ef-5a199ec6e8fe%2F9xv00yu_processed.png&w=3840&q=75)
Transcribed Image Text:**Solve the Differential Equation by Variation of Parameters**
Consider the differential equation:
\[ y'' + 3y' + 2y = \frac{1}{9 + e^x} \]
To solve this equation using the method of variation of parameters, follow these steps:
1. **Find the complementary solution (y_c):**
- Solve the homogeneous equation \( y'' + 3y' + 2y = 0 \).
2. **Particular solution (y_p) using variation of parameters:**
- Use the formula for variation of parameters to find a particular solution of the non-homogeneous equation.
3. **General Solution:**
- The general solution is \( y(x) = y_c(x) + y_p(x) \).
The solution will be displayed as:
\[ y(x) = \text{(general solution here)} \]
Complete the calculations to find the complete form of \( y(x) \).
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