ne point x = 0 is a regular singular point of the given differential equation. Find the recursive relation for the series solution of the DE below. Show the substitution and all the eps to obtain the recursive relation. Do not solve the equation for y=y(x) "+(1-x)y'-y=0
ne point x = 0 is a regular singular point of the given differential equation. Find the recursive relation for the series solution of the DE below. Show the substitution and all the eps to obtain the recursive relation. Do not solve the equation for y=y(x) "+(1-x)y'-y=0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Recursive Relation for Series Solution of a Differential Equation**
**Introduction:**
The point \( x = 0 \) is a regular singular point of the given differential equation. In this article, we will determine the recursive relation for the series solution of the differential equation provided below. This will include substitution and the detailed steps to obtain the recursive relation.
**Problem Statement:**
Find the recursive relation for the series solution of the following differential equation (DE). Show the substitution and all the steps to obtain the recursive relation. Note: Do not solve the equation for \( y = y(x) \).
\[ x y'' + (1 - x) y' - y = 0 \]
**Steps to Obtain the Recursive Relation:**
1. **Assume a Series Solution:**
Let's assume that the solution to the differential equation can be expressed as a power series around \( x = 0 \):
\[ y = \sum_{n=0}^{\infty} a_n x^n \]
2. **Compute the Derivatives:**
The first and second derivatives of the series solution are given by:
\[ y' = \sum_{n=1}^{\infty} n a_n x^{n-1} \]
\[ y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} \]
3. **Substitute into the Differential Equation:**
Substitute \( y \), \( y' \), and \( y'' \) into the given differential equation:
\[ x \left(\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}\right) + (1 - x) \left(\sum_{n=1}^{\infty} n a_n x^{n-1}\right) - \sum_{n=0}^{\infty} a_n x^n = 0 \]
4. **Simplify and Combine Series:**
Simplify the series expressions and combine terms to align coefficients of \( x^n \):
\[ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1} - \sum_{n=2}^{\infty} n(n-1) a_n x^n + \sum_{n=1}^{\in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9cdbf72-d206-4fdd-bded-1942fceb9e15%2Fb86f4233-638e-47ef-bff9-436e7ed1ab33%2F5l20ip8_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Recursive Relation for Series Solution of a Differential Equation**
**Introduction:**
The point \( x = 0 \) is a regular singular point of the given differential equation. In this article, we will determine the recursive relation for the series solution of the differential equation provided below. This will include substitution and the detailed steps to obtain the recursive relation.
**Problem Statement:**
Find the recursive relation for the series solution of the following differential equation (DE). Show the substitution and all the steps to obtain the recursive relation. Note: Do not solve the equation for \( y = y(x) \).
\[ x y'' + (1 - x) y' - y = 0 \]
**Steps to Obtain the Recursive Relation:**
1. **Assume a Series Solution:**
Let's assume that the solution to the differential equation can be expressed as a power series around \( x = 0 \):
\[ y = \sum_{n=0}^{\infty} a_n x^n \]
2. **Compute the Derivatives:**
The first and second derivatives of the series solution are given by:
\[ y' = \sum_{n=1}^{\infty} n a_n x^{n-1} \]
\[ y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} \]
3. **Substitute into the Differential Equation:**
Substitute \( y \), \( y' \), and \( y'' \) into the given differential equation:
\[ x \left(\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}\right) + (1 - x) \left(\sum_{n=1}^{\infty} n a_n x^{n-1}\right) - \sum_{n=0}^{\infty} a_n x^n = 0 \]
4. **Simplify and Combine Series:**
Simplify the series expressions and combine terms to align coefficients of \( x^n \):
\[ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1} - \sum_{n=2}^{\infty} n(n-1) a_n x^n + \sum_{n=1}^{\in
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