Consider the following differential equation to be solved using a power series. y" + xy = 0 00 Using the substitution y = cx, find an expression for Ck + 2 in terms of ck - 1 n=0 for k= 1, 2, 3....
Consider the following differential equation to be solved using a power series. y" + xy = 0 00 Using the substitution y = cx, find an expression for Ck + 2 in terms of ck - 1 n=0 for k= 1, 2, 3....
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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Question
Consider the following differential equation to be solved using a power series.
y'' + xy = 0
Using the substitution
y =
,
find an expression for
| ∞ | cnxn |
| n = 0 |
ck + 2
in terms of
ck − 1
for
k = 1, 2, 3 .
ck + 2 =
Ck−1(k+2)(k+1)
![**Differential Equation Power Series Solution**
**Problem Statement:**
Consider the following differential equation to be solved using a power series:
\[ y'' + xy = 0 \]
**Power Series Approach:**
Using the substitution:
\[ y = \sum_{n=0}^{\infty} c_n x^n \]
**Objective:**
Find an expression for \( c_{k+2} \) in terms of \( c_{k-1} \) for \( k = 1, 2, 3, \ldots \).
**Expression:**
\[ c_{k+2} = \frac{c_{k-1}}{(k+2)(k+1)} \]
**Additional Notes:**
In this context, the coefficients \( c_k \) of the power series solution are derived from the recurrence relation for solving the differential equation. The given equation shows the relationship between these coefficients for determining subsequent terms in the power series.
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Transcribed Image Text:**Differential Equation Power Series Solution**
**Problem Statement:**
Consider the following differential equation to be solved using a power series:
\[ y'' + xy = 0 \]
**Power Series Approach:**
Using the substitution:
\[ y = \sum_{n=0}^{\infty} c_n x^n \]
**Objective:**
Find an expression for \( c_{k+2} \) in terms of \( c_{k-1} \) for \( k = 1, 2, 3, \ldots \).
**Expression:**
\[ c_{k+2} = \frac{c_{k-1}}{(k+2)(k+1)} \]
**Additional Notes:**
In this context, the coefficients \( c_k \) of the power series solution are derived from the recurrence relation for solving the differential equation. The given equation shows the relationship between these coefficients for determining subsequent terms in the power series.
(There are no graphs or diagrams in the provided image that need further explanation.)
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