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
Related questions
Topic Video
Question
How can i solve this exercise?
![**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,
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)