Solve the differential equation by variation of parameters. 9x y(x) = y" - 9y = C₁e³r 3x 3x -3.x + C₂e -3.x + e (- 4 - 3r²) X
Solve the differential equation by variation of parameters. 9x y(x) = y" - 9y = C₁e³r 3x 3x -3.x + C₂e -3.x + e (- 4 - 3r²) X
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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![## Solving Differential Equations by Variation of Parameters
In this section, we will explore the method of solving a second-order linear differential equation using the variation of parameters. The differential equation given is:
\[ y'' - 9y = \frac{9x}{e^{3x}} \]
### Steps to Solve by Variation of Parameters
1. **Identify the Homogeneous Part**:
- The associated homogeneous equation is:
\[ y'' - 9y = 0 \]
2. **Solve the Homogeneous Equation**:
- The solution to the homogeneous equation can be found by solving the characteristic equation:
\[ r^2 - 9 = 0 \]
\[ r = \pm 3 \]
- Therefore, the general solution to the homogeneous equation is:
\[ y_h(x) = C_1 e^{3x} + C_2 e^{-3x} \]
3. **Particular Solution by Variation of Parameters**:
- Assume a particular solution of the form:
\[ y_p(x) = u_1(x) e^{3x} + u_2(x) e^{-3x} \]
- Use the method of variation of parameters to find \( u_1(x) \) and \( u_2(x) \).
4. **Construct the General Solution**:
- Combine the homogeneous solution and the particular solution to express the complete solution:
\[ y(x) = y_h(x) + y_p(x) \]
The general solution to the differential equation given is:
\[ y(x) = C_1 e^{3x} + C_2 e^{-3x} + e^{-3x} \left( -\frac{1}{4} - 3x^2 \right) \]
Where \( C_1 \) and \( C_2 \) are constants determined by initial conditions.
Note: The final expression inside the box represents the complete solution, highlighting both the homogeneous and particular solutions combined.
By solving step by step, students will gain a better understanding of applying variation of parameters and solving non-homogeneous differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21e8b9dc-8a46-44e8-b3f1-f32e58de6fd1%2Fb43ca690-bf37-4cfa-9784-fde9d8d2992d%2Fk3uny3t_processed.png&w=3840&q=75)
Transcribed Image Text:## Solving Differential Equations by Variation of Parameters
In this section, we will explore the method of solving a second-order linear differential equation using the variation of parameters. The differential equation given is:
\[ y'' - 9y = \frac{9x}{e^{3x}} \]
### Steps to Solve by Variation of Parameters
1. **Identify the Homogeneous Part**:
- The associated homogeneous equation is:
\[ y'' - 9y = 0 \]
2. **Solve the Homogeneous Equation**:
- The solution to the homogeneous equation can be found by solving the characteristic equation:
\[ r^2 - 9 = 0 \]
\[ r = \pm 3 \]
- Therefore, the general solution to the homogeneous equation is:
\[ y_h(x) = C_1 e^{3x} + C_2 e^{-3x} \]
3. **Particular Solution by Variation of Parameters**:
- Assume a particular solution of the form:
\[ y_p(x) = u_1(x) e^{3x} + u_2(x) e^{-3x} \]
- Use the method of variation of parameters to find \( u_1(x) \) and \( u_2(x) \).
4. **Construct the General Solution**:
- Combine the homogeneous solution and the particular solution to express the complete solution:
\[ y(x) = y_h(x) + y_p(x) \]
The general solution to the differential equation given is:
\[ y(x) = C_1 e^{3x} + C_2 e^{-3x} + e^{-3x} \left( -\frac{1}{4} - 3x^2 \right) \]
Where \( C_1 \) and \( C_2 \) are constants determined by initial conditions.
Note: The final expression inside the box represents the complete solution, highlighting both the homogeneous and particular solutions combined.
By solving step by step, students will gain a better understanding of applying variation of parameters and solving non-homogeneous differential equations.
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