Consider the following differential equation. 3x2y" + 2xy + 9x²y = 0 (c) Find the series solution (x > 0) corresponding to the larger root. O (-1)*9* y(x) = x 1 + Σk!-7-13---(6k + 1) O O O y(x) = x1/3 y(x) = x¹²21 + 1+ 00 y(x) = x¹|1 + [ [1 k=1 O O (-1)*3k x² y(x) = x/³1+k1-5-11---(3k + 1)2 k=1 y(x) = 1 + E k=1\ O eTextbook and Media O 00 y(x) = 1 + Y k=1 00 k1 k=1 00 y(x) = 1 + Y k=1 00 (d) Assuming the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also.. 00 k=1 (-1)*9k k!-5-11---(6k+ 1)2 k=1 (-1)k9k k!-7-13---(3k + 1) (-1)*2* Σk!-7-13--(6k+1) (-1)k9k k!-7-13---(3k + 1) (-1)k9k k!-7-11 (3k - 1) 2 00 y() = 1 + Σ. k=1 (1 ()] (-1)*3k k!-5-11---(6k+1) (-1)*9k y(x) = 1+k15-11---(6k-1 (-1)*9* k!-7-13---(6k-1) (3)] k k (³-³)* Assistance Used ()*
Consider the following differential equation. 3x2y" + 2xy + 9x²y = 0 (c) Find the series solution (x > 0) corresponding to the larger root. O (-1)*9* y(x) = x 1 + Σk!-7-13---(6k + 1) O O O y(x) = x1/3 y(x) = x¹²21 + 1+ 00 y(x) = x¹|1 + [ [1 k=1 O O (-1)*3k x² y(x) = x/³1+k1-5-11---(3k + 1)2 k=1 y(x) = 1 + E k=1\ O eTextbook and Media O 00 y(x) = 1 + Y k=1 00 k1 k=1 00 y(x) = 1 + Y k=1 00 (d) Assuming the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also.. 00 k=1 (-1)*9k k!-5-11---(6k+ 1)2 k=1 (-1)k9k k!-7-13---(3k + 1) (-1)*2* Σk!-7-13--(6k+1) (-1)k9k k!-7-13---(3k + 1) (-1)k9k k!-7-11 (3k - 1) 2 00 y() = 1 + Σ. k=1 (1 ()] (-1)*3k k!-5-11---(6k+1) (-1)*9k y(x) = 1+k15-11---(6k-1 (-1)*9* k!-7-13---(6k-1) (3)] k k (³-³)* Assistance Used ()*
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 Equation Problem
**Consider the following differential equation:**
\[ 3x^2y'' + 2xy' + 9x^2y = 0 \]
#### (c) Find the Series Solution (\( x > 0 \)) Corresponding to the Larger Root.
Which of the following provides the correct series solution corresponding to the larger root?
1. \[ y(x) = x^{1/3} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (6k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
2. \[ y(x) = x^{1/3} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 5 \cdot 11 \cdots (6k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
3. \[ y(x) = x^{1/2} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (3k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
4. \[ y(x) = x^{1/2} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (3k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
#### (d) Assuming the Roots are Unequal and Do Not Differ by an Integer, Find the Series Solution Corresponding to the Smaller Root Also.
Which of the following provides the correct series solution corresponding to the smaller root?
1. \[ y(x) = 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F8e419df2-2dec-4bfc-9ab9-4889ab46c88e%2F4gd7ga5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equation Problem
**Consider the following differential equation:**
\[ 3x^2y'' + 2xy' + 9x^2y = 0 \]
#### (c) Find the Series Solution (\( x > 0 \)) Corresponding to the Larger Root.
Which of the following provides the correct series solution corresponding to the larger root?
1. \[ y(x) = x^{1/3} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (6k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
2. \[ y(x) = x^{1/3} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 5 \cdot 11 \cdots (6k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
3. \[ y(x) = x^{1/2} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (3k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
4. \[ y(x) = x^{1/2} \left[ 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot 7 \cdot 13 \cdots (3k + 1)} \left( \frac{x^2}{2} \right)^k \right] \]
#### (d) Assuming the Roots are Unequal and Do Not Differ by an Integer, Find the Series Solution Corresponding to the Smaller Root Also.
Which of the following provides the correct series solution corresponding to the smaller root?
1. \[ y(x) = 1 + \sum_{k=1}^{\infty} \frac{(-1)^k 9^k}{k! \cdot
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