Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Recursive Relation for Series Solution of a Differential Equation**
**Problem Statement:**
The point \( x = 0 \) is a regular singular point of the given differential equation. **Find the recursive relation for the series solution of the differential equation (DE) below.** Show the substitution and all steps to obtain the recursive relation. **Do not solve the equation for \( y = y(x) \).**
**Differential Equation:**
\[ xy'' + 4y' - xy = 0, \]
**Potential Recursive Relations:**
a. \( c_{k+1} = \frac{1}{(k + r + 1)(k + r + 3)}c_{k-1}, \quad k \geq 1 \)
b. \( c_{k+1} = \frac{1}{(k + r + 1)(k + r + 4)}c_{k-1}, \quad k \geq 1 \)
c. \( c_k = \frac{1}{k + r}c_{k-1}, \quad k \geq 1 \)
d. \( c_k = -\frac{k + r}{(k + r)^2 + 5(k + r)}c_{k-1}, \quad k \geq 1 \)
e. \( c_k = -\frac{1}{(k + r)^2 - 2(k + r) - 8}c_{k-2}, \quad k \geq 2 \)
**Explanation:**
This exercise involves verifying which option provides the correct recursive relation for the coefficients \( c_k \) in the series solution to the differential equation. To solve it, you would typically:
1. Assume a power series solution \( y = \sum_{k=0}^{\infty} c_k x^{k+r} \).
2. Substitute this into the differential equation: \[ x \sum_{k=0}^{\infty} (k+r)(k+r-1)c_k x^{k+r-2} + 4 \sum_{k=0}^{\infty} (k+r)c_k x^{k+r-1} - x \sum_{k=0}^{\infty} c_k x^{k+r} = 0. \]
3. Combine and simplify the series terms.
4. Obtain a recurrence relation](https://content.bartleby.com/qna-images/question/46272ace-8a35-41ec-9b9a-0375bd37bb55/7269c82e-e55a-4c6c-a854-b44d132be6d3/kvr0qlm_processed.png)
Transcribed Image Text:**Title: Recursive Relation for Series Solution of a Differential Equation**
**Problem Statement:**
The point \( x = 0 \) is a regular singular point of the given differential equation. **Find the recursive relation for the series solution of the differential equation (DE) below.** Show the substitution and all steps to obtain the recursive relation. **Do not solve the equation for \( y = y(x) \).**
**Differential Equation:**
\[ xy'' + 4y' - xy = 0, \]
**Potential Recursive Relations:**
a. \( c_{k+1} = \frac{1}{(k + r + 1)(k + r + 3)}c_{k-1}, \quad k \geq 1 \)
b. \( c_{k+1} = \frac{1}{(k + r + 1)(k + r + 4)}c_{k-1}, \quad k \geq 1 \)
c. \( c_k = \frac{1}{k + r}c_{k-1}, \quad k \geq 1 \)
d. \( c_k = -\frac{k + r}{(k + r)^2 + 5(k + r)}c_{k-1}, \quad k \geq 1 \)
e. \( c_k = -\frac{1}{(k + r)^2 - 2(k + r) - 8}c_{k-2}, \quad k \geq 2 \)
**Explanation:**
This exercise involves verifying which option provides the correct recursive relation for the coefficients \( c_k \) in the series solution to the differential equation. To solve it, you would typically:
1. Assume a power series solution \( y = \sum_{k=0}^{\infty} c_k x^{k+r} \).
2. Substitute this into the differential equation: \[ x \sum_{k=0}^{\infty} (k+r)(k+r-1)c_k x^{k+r-2} + 4 \sum_{k=0}^{\infty} (k+r)c_k x^{k+r-1} - x \sum_{k=0}^{\infty} c_k x^{k+r} = 0. \]
3. Combine and simplify the series terms.
4. Obtain a recurrence relation
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