4. Use the series method to fin dthe first five non-zero terms of the solutio of the following IVP: with y(0) = 1, y'(0) = 3. y" + xy = 2x
4. Use the series method to fin dthe first five non-zero terms of the solutio of the following IVP: with y(0) = 1, y'(0) = 3. y" + xy = 2x
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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![**Series Method for Solving Differential Equations**
In this exercise, we are tasked with using the series method to find the first five non-zero terms of the solution to the following Initial Value Problem (IVP):
\[ y'' + xy = 2x \]
Subject to the initial conditions:
\[ y(0) = 1, \quad y'(0) = 3. \]
**Objective:**
- Determine the first five non-zero terms of the solution \( y(x) \) using the series method.
**Steps for Solving Using Series Method:**
1. **Assume a Power Series Solution:**
We begin by assuming that the solution can be expressed as a power series:
\[ y(x) = \sum_{n=0}^{\infty} a_n x^n. \]
2. **Compute Derivatives:**
Find the first and second derivatives of \( y(x) \):
\[ y'(x) = \sum_{n=1}^{\infty} n a_n x^{n - 1}, \]
\[ y''(x) = \sum_{n=2}^{\infty} n (n - 1) a_n x^{n - 2}. \]
3. **Substitute into Differential Equation:**
Substitute \( y(x) \) and its derivatives into the given differential equation to obtain an equation involving the series:
\[ \sum_{n=2}^{\infty} n (n - 1) a_n x^{n - 2} + x \sum_{n=0}^{\infty} a_n x^n = 2x. \]
4. **Equate Coefficients:**
After simplification, equate the coefficients of powers of \( x \) on both sides of the equation to find a recurrence relation for the coefficients \( a_n \).
5. **Find Initial Terms:**
Use the initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \) to find the values of \( a_0 \) and \( a_1 \), and then determine the subsequent terms \( a_2, a_3, a_4, \) and \( a_5 \).
This approach will yield the first five non-zero terms of the power series solution to the initial value problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53b47ed7-4497-4fb9-9d0f-c41504626a91%2F8c6373b4-33cc-4837-8f89-153b3cc1acbe%2Fxogzsvg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Series Method for Solving Differential Equations**
In this exercise, we are tasked with using the series method to find the first five non-zero terms of the solution to the following Initial Value Problem (IVP):
\[ y'' + xy = 2x \]
Subject to the initial conditions:
\[ y(0) = 1, \quad y'(0) = 3. \]
**Objective:**
- Determine the first five non-zero terms of the solution \( y(x) \) using the series method.
**Steps for Solving Using Series Method:**
1. **Assume a Power Series Solution:**
We begin by assuming that the solution can be expressed as a power series:
\[ y(x) = \sum_{n=0}^{\infty} a_n x^n. \]
2. **Compute Derivatives:**
Find the first and second derivatives of \( y(x) \):
\[ y'(x) = \sum_{n=1}^{\infty} n a_n x^{n - 1}, \]
\[ y''(x) = \sum_{n=2}^{\infty} n (n - 1) a_n x^{n - 2}. \]
3. **Substitute into Differential Equation:**
Substitute \( y(x) \) and its derivatives into the given differential equation to obtain an equation involving the series:
\[ \sum_{n=2}^{\infty} n (n - 1) a_n x^{n - 2} + x \sum_{n=0}^{\infty} a_n x^n = 2x. \]
4. **Equate Coefficients:**
After simplification, equate the coefficients of powers of \( x \) on both sides of the equation to find a recurrence relation for the coefficients \( a_n \).
5. **Find Initial Terms:**
Use the initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \) to find the values of \( a_0 \) and \( a_1 \), and then determine the subsequent terms \( a_2, a_3, a_4, \) and \( a_5 \).
This approach will yield the first five non-zero terms of the power series solution to the initial value problem.
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