Use power series to solve the initial-value problem Answer: y = ∞0 n=0 ²n + 8 72=() y" + 3xy' + 3y = 0, x2n+1 y(0) = 1, y' (0) = 0
Use power series to solve the initial-value problem Answer: y = ∞0 n=0 ²n + 8 72=() y" + 3xy' + 3y = 0, x2n+1 y(0) = 1, y' (0) = 0
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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![**Using Power Series to Solve the Initial-Value Problem**
We are tasked with solving the following initial-value problem using a power series approach:
\[ y'' + 3xy' + 3y = 0, \]
\[ y(0) = 1, \]
\[ y'(0) = 0. \]
**Solution:**
We express the solution \( y \) as a power series:
\[ y = \sum_{n=0}^{\infty} a_n x^n + x^2 \sum_{n=0}^{\infty} a_n x^{2n+1}. \]
This form allows for solving the differential equation and applying the given initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91caee7e-a450-4823-b604-91c802978c64%2F8f22f86b-5759-44e8-a29d-27d4b69f8b29%2F7t0uvkg_processed.png&w=3840&q=75)
Transcribed Image Text:**Using Power Series to Solve the Initial-Value Problem**
We are tasked with solving the following initial-value problem using a power series approach:
\[ y'' + 3xy' + 3y = 0, \]
\[ y(0) = 1, \]
\[ y'(0) = 0. \]
**Solution:**
We express the solution \( y \) as a power series:
\[ y = \sum_{n=0}^{\infty} a_n x^n + x^2 \sum_{n=0}^{\infty} a_n x^{2n+1}. \]
This form allows for solving the differential equation and applying the given initial conditions.
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