Consider the following differential equation to be solved by variation of parameters. 4y" − y = ex/2 +6 Find the complementary function of the differential equation. ret) - G₂e (3) + eye (3) yc(x) = Find the general solution of the differential equation. y(x) = - (3) + €₂0 (³) — < (9) + x²( ³ ) — 24 e( e xe е C₂² c₁e X
Consider the following differential equation to be solved by variation of parameters. 4y" − y = ex/2 +6 Find the complementary function of the differential equation. ret) - G₂e (3) + eye (3) yc(x) = Find the general solution of the differential equation. y(x) = - (3) + €₂0 (³) — < (9) + x²( ³ ) — 24 e( e xe е C₂² c₁e 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
Consider the following differential equation to be solved by variation of parameters:
\[ 4y'' - y = e^{x/2} + 6 \]
#### Step 1: Find the Complementary Function of the Differential Equation
The complementary function \( y_c(x) \) is given by:
\[ y_c(x) = c_1 e^{\left(\frac{x}{2}\right)} + c_2 e^{\left(-\frac{x}{2}\right)} \]
*Note: The green checkmark indicates that this solution is verified.*
#### Step 2: Find the General Solution of the Differential Equation
The general solution \( y(x) \) can be written as:
\[ y(x) = c_1 e^{\left(-\frac{x}{2}\right)} + c_2 e^{\left(\frac{x}{2}\right)} - e^{\left(\frac{x}{2}\right)} + x e^{\left(\frac{x}{2}\right)} - 24 \]
*Note: The red cross indicates a mistake or error in this proposed solution.*
---
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There are no specific diagrams or graphs in this example to explain. The provided equations and solutions are algebraic representations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96023f4d-6281-4c81-803e-adfe3e4197d6%2Fe70a63d4-c33b-4753-ab00-c1b69f0411ac%2Fwgddzsc_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations by Variation of Parameters
Consider the following differential equation to be solved by variation of parameters:
\[ 4y'' - y = e^{x/2} + 6 \]
#### Step 1: Find the Complementary Function of the Differential Equation
The complementary function \( y_c(x) \) is given by:
\[ y_c(x) = c_1 e^{\left(\frac{x}{2}\right)} + c_2 e^{\left(-\frac{x}{2}\right)} \]
*Note: The green checkmark indicates that this solution is verified.*
#### Step 2: Find the General Solution of the Differential Equation
The general solution \( y(x) \) can be written as:
\[ y(x) = c_1 e^{\left(-\frac{x}{2}\right)} + c_2 e^{\left(\frac{x}{2}\right)} - e^{\left(\frac{x}{2}\right)} + x e^{\left(\frac{x}{2}\right)} - 24 \]
*Note: The red cross indicates a mistake or error in this proposed solution.*
---
### Need Help?
Click the "Read It" button for additional guidance and explanation.
### Interactive Features
- **Submit Answer**: Allows you to submit your answer for evaluation.
- **Feedback Icons**:
- ✅ Green checkmark indicates a correct step or result.
- ❌ Red cross suggests an incorrect or erroneous response.
### Diagram and Graph Explanation
There are no specific diagrams or graphs in this example to explain. The provided equations and solutions are algebraic representations.
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