The 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 steps to obtain the recursive relation. Do not solve the equation for y=y(x) xy" + 4y' - xy = 0,

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
### Differential Equations: Series Solutions at Regular Singular Points

The point \( x = 0 \) is a regular singular point of the given differential equation. In this exercise, we will **find the recursive relation for the series solution of the differential equation (DE) below**. Follow the steps to show the substitution and all the methods used to obtain the recursive relation. Note: We will not solve the equation for \( y = y(x) \) at this stage.

The given differential equation is:
\[ xy'' + 4y' - xy = 0, \]

To find the series solution, assume that the solution can be expressed in the form of a power series:
\[ y = \sum_{n=0}^{\infty} a_n x^n. \]

The first and second derivatives of \( y \) are:
\[ y' = \sum_{n=1}^{\infty} n a_n x^{n-1}, \]
\[ y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}. \]

Now, substitute \( y \), \( y' \), and \( y'' \) back into the original differential equation and obtain a recursive relation for the coefficients \( a_n \).

1. **Substitute \( y'' \) into the equation:**
\[ xy'' = x \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1}. \]

2. **Substitute \( y' \) into the equation:**
\[ 4y' = 4 \sum_{n=1}^{\infty} n a_n x^{n-1}. \]

3. **Substitute \( y \) into the equation:**
\[ -xy = -x \sum_{n=0}^{\infty} a_n x^n = -\sum_{n=0}^{\infty} a_n x^{n+1}. \]

Combine these results to get:
\[ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1} + 4 \sum_{n=1}^{\infty} n a_n x^{n-1} - \sum_{n
Transcribed Image Text:### Differential Equations: Series Solutions at Regular Singular Points The point \( x = 0 \) is a regular singular point of the given differential equation. In this exercise, we will **find the recursive relation for the series solution of the differential equation (DE) below**. Follow the steps to show the substitution and all the methods used to obtain the recursive relation. Note: We will not solve the equation for \( y = y(x) \) at this stage. The given differential equation is: \[ xy'' + 4y' - xy = 0, \] To find the series solution, assume that the solution can be expressed in the form of a power series: \[ y = \sum_{n=0}^{\infty} a_n x^n. \] The first and second derivatives of \( y \) are: \[ y' = \sum_{n=1}^{\infty} n a_n x^{n-1}, \] \[ y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}. \] Now, substitute \( y \), \( y' \), and \( y'' \) back into the original differential equation and obtain a recursive relation for the coefficients \( a_n \). 1. **Substitute \( y'' \) into the equation:** \[ xy'' = x \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1}. \] 2. **Substitute \( y' \) into the equation:** \[ 4y' = 4 \sum_{n=1}^{\infty} n a_n x^{n-1}. \] 3. **Substitute \( y \) into the equation:** \[ -xy = -x \sum_{n=0}^{\infty} a_n x^n = -\sum_{n=0}^{\infty} a_n x^{n+1}. \] Combine these results to get: \[ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-1} + 4 \sum_{n=1}^{\infty} n a_n x^{n-1} - \sum_{n
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,