The point x = 0 is a regular singular point of the given differential equation. 4xy" y' + 4y = 0 Show that the indicial roots r of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list.) r = Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, co). Oy=(1-x++) + ₂x² (₁-x²) C₂x5/4 (1- 4 4 36 Oy=c₁(1 = C₁ 1- 4x + Oy=C₁x5/4 Oy=c₁₂ (1+ 1+ 4x 8x2 32x3 5 135 36 1+ 4x - - 8x² 8x² 32x³ 3 63 + + 5 + 32x³ 63 ..) + + 1-x+ 2+5/4(1 +... + C₂x³/ + 1- 1- ..) + C₂x5/4(1- 4x 8x² 7 77 +.. + 4x 9 4x 8x² + 9 117 8x2 32x3 117 5967 Oy=c₁²/1-4x+8²-32x³ + ...) + C₂(1 - 4x + 8x² - 32x²- (1 7 77 3465 32x3 3465 32x3 5967
The point x = 0 is a regular singular point of the given differential equation. 4xy" y' + 4y = 0 Show that the indicial roots r of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list.) r = Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, co). Oy=(1-x++) + ₂x² (₁-x²) C₂x5/4 (1- 4 4 36 Oy=c₁(1 = C₁ 1- 4x + Oy=C₁x5/4 Oy=c₁₂ (1+ 1+ 4x 8x2 32x3 5 135 36 1+ 4x - - 8x² 8x² 32x³ 3 63 + + 5 + 32x³ 63 ..) + + 1-x+ 2+5/4(1 +... + C₂x³/ + 1- 1- ..) + C₂x5/4(1- 4x 8x² 7 77 +.. + 4x 9 4x 8x² + 9 117 8x2 32x3 117 5967 Oy=c₁²/1-4x+8²-32x³ + ...) + C₂(1 - 4x + 8x² - 32x²- (1 7 77 3465 32x3 3465 32x3 5967
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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![### Singular Points and Series Solutions for Differential Equations Using the Frobenius Method
#### Problem Statement
The point \( x = 0 \) is a regular singular point of the given differential equation:
\[ 4xy'' - y' + 4y = 0 \]
#### Task 1: Indicial Roots
Show that the indicial roots \( r \) of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list).
\[ r = \_\_\_\_\_\_ \]
#### Task 2: Frobenius Method
Use the method of Frobenius to obtain two linearly independent series solutions about \( x = 0 \). Form the general solution on \( (0, \infty) \).
#### Options for the General Solution
1.
\[ y = C_1 \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) + C_2 x^{5/4} \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) \]
2.
\[ y = C_1 \left( 1 - 4x + \frac{8x^2}{5} - \frac{32x^3}{135} + \cdots \right) + C_2 x^{5/4} \left( 1 - \frac{4x}{7} + \frac{8x^2}{77} - \frac{32x^3}{3465} + \cdots \right) \]
3.
\[ y = C_1 x^{5/4} \left( 1 + 4x - \frac{8x^2}{3} + \frac{32x^3}{63} + \cdots \right) + C_2 \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) \]
4.
\[ y = C_1 \left( 1 + 4x - \frac{8x^2}{3} + \frac{32x^3}{63} + \cdots \right) + C_2 x^{5/4}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F995b5b40-26b1-420f-ae4c-d7493773d7f9%2F5af8c08b-1606-44d6-a375-32e6d9d1001d%2Fp7nzz0q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Singular Points and Series Solutions for Differential Equations Using the Frobenius Method
#### Problem Statement
The point \( x = 0 \) is a regular singular point of the given differential equation:
\[ 4xy'' - y' + 4y = 0 \]
#### Task 1: Indicial Roots
Show that the indicial roots \( r \) of the singularity do not differ by an integer. (List the indicial roots below as a comma-separated list).
\[ r = \_\_\_\_\_\_ \]
#### Task 2: Frobenius Method
Use the method of Frobenius to obtain two linearly independent series solutions about \( x = 0 \). Form the general solution on \( (0, \infty) \).
#### Options for the General Solution
1.
\[ y = C_1 \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) + C_2 x^{5/4} \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) \]
2.
\[ y = C_1 \left( 1 - 4x + \frac{8x^2}{5} - \frac{32x^3}{135} + \cdots \right) + C_2 x^{5/4} \left( 1 - \frac{4x}{7} + \frac{8x^2}{77} - \frac{32x^3}{3465} + \cdots \right) \]
3.
\[ y = C_1 x^{5/4} \left( 1 + 4x - \frac{8x^2}{3} + \frac{32x^3}{63} + \cdots \right) + C_2 \left( 1 - x + \frac{x^2}{4} - \frac{x^3}{36} + \cdots \right) \]
4.
\[ y = C_1 \left( 1 + 4x - \frac{8x^2}{3} + \frac{32x^3}{63} + \cdots \right) + C_2 x^{5/4}
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