Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (Use D for the differential operator.) 4y" – 15y' – 4y = 5 = 5
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (Use D for the differential operator.) 4y" – 15y' – 4y = 5 = 5
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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Question
![**Title: Converting Differential Equations to Operator Form**
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**Objective:**
Convert the differential equation into the form \( L(y) = g(x) \), where \( L \) is a linear differential operator with constant coefficients. Factor \( L \) if possible. Use \( D \) for the differential operator.
**Given Equation:**
\[ 4y'' - 15y' - 4y = 5 \]
**Form Conversion:**
\[ ( \underline{\quad\quad\quad} ) y = 5 \]
In this task, you need to express the differential equation by applying the differential operator \( D \), where \( D = \frac{d}{dx} \). This involves transforming the equation into an operator format and attempting to factor it if possible.
**Explanation:**
1. **Identify Components:**
- \( y'' = D^2y \)
- \( y' = Dy \)
- \( y = y \)
2. **Operator Reformation:**
- Substitute these into the equation to form the operator expression:
\[ 4D^2y - 15Dy - 4y = 5 \]
3. **Factoring:**
- This operator \( L = 4D^2 - 15D - 4 \) may be factored into simpler differential operators if possible.
**Note:** This helps in simplifying and solving differential equations systematically using linear algebra techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd39804f-6d08-4135-954a-c92ac4c2d043%2F81a24a2a-1043-49ca-8c26-e6451bcdca4f%2F1c69lmr_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Converting Differential Equations to Operator Form**
---
**Objective:**
Convert the differential equation into the form \( L(y) = g(x) \), where \( L \) is a linear differential operator with constant coefficients. Factor \( L \) if possible. Use \( D \) for the differential operator.
**Given Equation:**
\[ 4y'' - 15y' - 4y = 5 \]
**Form Conversion:**
\[ ( \underline{\quad\quad\quad} ) y = 5 \]
In this task, you need to express the differential equation by applying the differential operator \( D \), where \( D = \frac{d}{dx} \). This involves transforming the equation into an operator format and attempting to factor it if possible.
**Explanation:**
1. **Identify Components:**
- \( y'' = D^2y \)
- \( y' = Dy \)
- \( y = y \)
2. **Operator Reformation:**
- Substitute these into the equation to form the operator expression:
\[ 4D^2y - 15Dy - 4y = 5 \]
3. **Factoring:**
- This operator \( L = 4D^2 - 15D - 4 \) may be factored into simpler differential operators if possible.
**Note:** This helps in simplifying and solving differential equations systematically using linear algebra techniques.
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