Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Differential Equations: Regular Singular Point and Indicial Equation
**Problem Statement:**
Show that the given differential equation
\[ 3x^2 y'' + \left( 2x^2 + \frac{3}{4} \right) y = 0 \]
has a regular singular point at \( x = 0 \).
**Steps to Solve:**
1. **Identify \( xp(x) \) and \( x^2 q(x) \):**
- \( xp(x) = \boxed{\hspace{3cm}} \)
- \( x^2 q(x) = \boxed{\hspace{3cm}} \)
2. **Find Indicial Equation:**
- The indicial equation is derived from the coefficients of the terms in the differential equation.
- Establish the leading coefficients for the series solutions around \( x = 0 \).
- \( 0 = \boxed{\hspace{3cm}} \)
3. **Determine the Number of Roots:**
- How many roots does the indicial equation have?
-](https://content.bartleby.com/qna-images/question/20c5837f-8a07-4fb0-b110-a8d2b8a51a74/7cff79a7-501f-4d6d-8076-f5483b391e77/1o2bgz_processed.jpeg)
Transcribed Image Text:### Differential Equations: Regular Singular Point and Indicial Equation
**Problem Statement:**
Show that the given differential equation
\[ 3x^2 y'' + \left( 2x^2 + \frac{3}{4} \right) y = 0 \]
has a regular singular point at \( x = 0 \).
**Steps to Solve:**
1. **Identify \( xp(x) \) and \( x^2 q(x) \):**
- \( xp(x) = \boxed{\hspace{3cm}} \)
- \( x^2 q(x) = \boxed{\hspace{3cm}} \)
2. **Find Indicial Equation:**
- The indicial equation is derived from the coefficients of the terms in the differential equation.
- Establish the leading coefficients for the series solutions around \( x = 0 \).
- \( 0 = \boxed{\hspace{3cm}} \)
3. **Determine the Number of Roots:**
- How many roots does the indicial equation have?
-
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