Find the power series solution for the differential equation y" + 3xy'-y=0 with the initial conditions y (0) = 2 and y'(0) = 0. A y=x²+. (U) 5 12 X 4 + 11 72 X 6 + 11 ® y=x²- 5 2x² + 1/2x²+... B 12 72 5 11 =2+x² + 2x² + 1/2 x 6+ ... 12 72 5 11 4 Ⓒy=2+x² - 1/2x² + 1/2xX6 + ...
Find the power series solution for the differential equation y" + 3xy'-y=0 with the initial conditions y (0) = 2 and y'(0) = 0. A y=x²+. (U) 5 12 X 4 + 11 72 X 6 + 11 ® y=x²- 5 2x² + 1/2x²+... B 12 72 5 11 =2+x² + 2x² + 1/2 x 6+ ... 12 72 5 11 4 Ⓒy=2+x² - 1/2x² + 1/2xX6 + ...
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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![### Power Series Solution for Differential Equation
**Problem Statement:**
Find the power series solution for the differential equation \( y'' + 3y' - y = 0 \) with the initial conditions \( y(0) = 2 \) and \( y'(0) = 0 \).
**Solution Options:**
A) \( y = x + \frac{5}{12} x^3 + \frac{11}{72} x^5 + \cdots \)
B) \( y = x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
C) \( y = 2 + x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
D) \( y = 2 + x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
### Power Series Detailed Explanation:
The differential equation \( y'' + 3y' - y = 0 \) can be solved using a power series approach. A power series solution is typically expressed in the form:
\[ y(x) = \sum_{n=0}^{\infty} a_n x^n \]
Based on the given initial conditions and the specific problem requirements, each option provides a different potential series expansion. Let's briefly analyze how to arrive at the correct power series solution:
1. **Initial Conditions:**
- \( y(0) = 2 \) suggests that the constant term \( a_0 = 2 \).
- \( y'(0) = 0 \) implies that the coefficient \( a_1 = 0 \).
2. **Typical Power Series Terms:**
For the given differential equation, substitute the power series into the equation and match coefficients to find the correct expansion:
\[ y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots \]
Given the initial conditions:
\[ y(0) = a_0 = 2 \]
\[ y'(0) = a_1 = 0 \]
By updating the coefficients through mathematical iterations and substitutions, we find the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dd05a02-deea-4dc4-8733-d9131927e1e8%2Fcce33d88-53ab-4424-81d6-8f33bca4ecd4%2Fz4jys3w_processed.png&w=3840&q=75)
Transcribed Image Text:### Power Series Solution for Differential Equation
**Problem Statement:**
Find the power series solution for the differential equation \( y'' + 3y' - y = 0 \) with the initial conditions \( y(0) = 2 \) and \( y'(0) = 0 \).
**Solution Options:**
A) \( y = x + \frac{5}{12} x^3 + \frac{11}{72} x^5 + \cdots \)
B) \( y = x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
C) \( y = 2 + x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
D) \( y = 2 + x^2 - \frac{5}{12} x^4 + \frac{11}{72} x^6 + \cdots \)
### Power Series Detailed Explanation:
The differential equation \( y'' + 3y' - y = 0 \) can be solved using a power series approach. A power series solution is typically expressed in the form:
\[ y(x) = \sum_{n=0}^{\infty} a_n x^n \]
Based on the given initial conditions and the specific problem requirements, each option provides a different potential series expansion. Let's briefly analyze how to arrive at the correct power series solution:
1. **Initial Conditions:**
- \( y(0) = 2 \) suggests that the constant term \( a_0 = 2 \).
- \( y'(0) = 0 \) implies that the coefficient \( a_1 = 0 \).
2. **Typical Power Series Terms:**
For the given differential equation, substitute the power series into the equation and match coefficients to find the correct expansion:
\[ y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots \]
Given the initial conditions:
\[ y(0) = a_0 = 2 \]
\[ y'(0) = a_1 = 0 \]
By updating the coefficients through mathematical iterations and substitutions, we find the
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