Find at least the first four nonzero terms in a power series expansion about x, for a genera solution to the given Differential equation with the given value for x, y" +(3x–1)y – y=0, xo =-1
Find at least the first four nonzero terms in a power series expansion about x, for a genera solution to the given Differential equation with the given value for x, y" +(3x–1)y – y=0, xo =-1
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 Expansion for Differential Equations**
In this section, we will explore how to find at least the first four nonzero terms in a power series expansion about \(x_0\) for a general solution to a given differential equation, given the specified value for \(x_0\).
Consider the differential equation given by:
\[ y'' + (3x - 1) y' - y = 0 \]
with the initial point \( x_0 = -1 \).
To solve this differential equation using a power series expansion, we follow these steps:
1. **Assume a Power Series Solution**:
Assume that the solution \(y(x)\) can be expressed as a power series centered at \(x_0 = -1\):
\[
y(x) = \sum_{n=0}^{\infty} a_n (x + 1)^n
\]
2. **Differentiating the Series**:
To find \(y'(x)\) and \(y''(x)\), differentiate the power series term by term:
\[
y'(x) = \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1}
\]
\[
y''(x) = \sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2}
\]
3. **Substituting into the Differential Equation**:
Substitute \(y(x)\), \(y'(x)\), and \(y''(x)\) back into the differential equation:
\[
\sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2} + (3x - 1) \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1} - \sum_{n=0}^{\infty} a_n (x + 1)^n = 0
\]
4. **Combine Like Terms and Solve for Coefficients**:
Align the powers of \((x + 1)\) and combine like terms. Setting coefficients of like powers of \((x + 1)\) to zero gives a recurrence relation for the coefficients \(a_n\).
From this,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6a44fb-472d-4115-ae99-90593ad92ba0%2Fdb0835f8-7148-4d3f-81aa-2e33f3780aef%2Fgdydej_processed.png&w=3840&q=75)
Transcribed Image Text:**Power Series Expansion for Differential Equations**
In this section, we will explore how to find at least the first four nonzero terms in a power series expansion about \(x_0\) for a general solution to a given differential equation, given the specified value for \(x_0\).
Consider the differential equation given by:
\[ y'' + (3x - 1) y' - y = 0 \]
with the initial point \( x_0 = -1 \).
To solve this differential equation using a power series expansion, we follow these steps:
1. **Assume a Power Series Solution**:
Assume that the solution \(y(x)\) can be expressed as a power series centered at \(x_0 = -1\):
\[
y(x) = \sum_{n=0}^{\infty} a_n (x + 1)^n
\]
2. **Differentiating the Series**:
To find \(y'(x)\) and \(y''(x)\), differentiate the power series term by term:
\[
y'(x) = \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1}
\]
\[
y''(x) = \sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2}
\]
3. **Substituting into the Differential Equation**:
Substitute \(y(x)\), \(y'(x)\), and \(y''(x)\) back into the differential equation:
\[
\sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2} + (3x - 1) \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1} - \sum_{n=0}^{\infty} a_n (x + 1)^n = 0
\]
4. **Combine Like Terms and Solve for Coefficients**:
Align the powers of \((x + 1)\) and combine like terms. Setting coefficients of like powers of \((x + 1)\) to zero gives a recurrence relation for the coefficients \(a_n\).
From this,
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