Show that the given differential equation 3x²y" + (2x² + 22² + ³) y y=0 has a regular singular point at x = 0. xp(x) = Find indicial equation. 0= Find the recurrence equation. x²q(x) = How many roots does the indicial equation have? Choose one ▾ an = 314 Choose one Choose one •An-1 for n = 1, 2, ... an-2 for n = 2, 3, ... for n-34
Show that the given differential equation 3x²y" + (2x² + 22² + ³) y y=0 has a regular singular point at x = 0. xp(x) = Find indicial equation. 0= Find the recurrence equation. x²q(x) = How many roots does the indicial equation have? Choose one ▾ an = 314 Choose one Choose one •An-1 for n = 1, 2, ... an-2 for n = 2, 3, ... for n-34
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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![### 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?
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F7cff79a7-501f-4d6d-8076-f5483b391e77%2F1o2bgz_processed.jpeg&w=3840&q=75)
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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