Solve the recurrence relation , if n = 0 if n = 1 ): N → R a := 1 (5ап-1 — бап-2 + 4n, if n > 1)
Solve the recurrence relation , if n = 0 if n = 1 ): N → R a := 1 (5ап-1 — бап-2 + 4n, if n > 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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Please help me solve this one. I need your help. I have attached the lecture slides too just in case. HELP ME PLEASE
![**Problem Statement: Solve the Recurrence Relation**
Define the sequence \( \alpha \) as follows:
\[
\alpha := \left( n \mapsto
\begin{cases}
0, & \text{if } n = 0 \\
1, & \text{if } n = 1 \\
5\alpha_{n-1} - 6\alpha_{n-2} + 4n, & \text{if } n > 1
\end{cases}
\right) : \mathbb{N} \to \mathbb{R}
\]
### Explanation:
You are required to find the solution for the recurrence relation defined above.
1. **Base Cases:**
- For \( n = 0 \), \( \alpha_n = 0 \).
- For \( n = 1 \), \( \alpha_n = 1 \).
2. **Recursive Case:**
- For \( n > 1 \), the relation is given by:
\[
\alpha_n = 5\alpha_{n-1} - 6\alpha_{n-2} + 4n
\]
Your task is to determine the explicit form of the sequence \( \alpha \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004a8e3c-a278-4414-bbaf-77c8d726d7bd%2Fad8ef7ab-d8e3-4345-ba1c-c6e00f6a0ff3%2Fcpyr6ek_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Solve the Recurrence Relation**
Define the sequence \( \alpha \) as follows:
\[
\alpha := \left( n \mapsto
\begin{cases}
0, & \text{if } n = 0 \\
1, & \text{if } n = 1 \\
5\alpha_{n-1} - 6\alpha_{n-2} + 4n, & \text{if } n > 1
\end{cases}
\right) : \mathbb{N} \to \mathbb{R}
\]
### Explanation:
You are required to find the solution for the recurrence relation defined above.
1. **Base Cases:**
- For \( n = 0 \), \( \alpha_n = 0 \).
- For \( n = 1 \), \( \alpha_n = 1 \).
2. **Recursive Case:**
- For \( n > 1 \), the relation is given by:
\[
\alpha_n = 5\alpha_{n-1} - 6\alpha_{n-2} + 4n
\]
Your task is to determine the explicit form of the sequence \( \alpha \).
![## Topic 2.6.0: Recurrence Relations
### Definition
Suppose \( k \in \mathbb{N} \) and \( f \) is a sequence function of the form:
\[
\begin{bmatrix}
x_0 \\
x_{n-1} \\
\vdots \\
x_{n-k+1}
\end{bmatrix}
\mapsto x_n = f(x_{n-1}, \ldots, x_{n-k})
\]
It is called a recurrence relation of order \( k \). If it has initial conditions such as:
\[
x_0 = a_0, \ldots, x_{k-1} = a_{k-1}
\]
Then the recurrence relation is of **order** \( k \).
### Examples include:
1. **Geometric Progression (GP)**: \( x_n = rx_{n-1} \)
2. **The Tower of Hanoi Sequence**: \( x_n = 2x_{n-1} + 1 \)
3. **The Fibonacci Sequence**: \( x_n = x_{n-1} + x_{n-2} \)
---
## Topic 2.6.1: Characteristic Equations
### Example
Suppose \( k \in \mathbb{N} \) and it's a linear recurrence relation of order \( k \):
\[
\begin{bmatrix}
x_0 \\
x_{n-1} \\
\vdots \\
x_{n-k+1}
\end{bmatrix}
\mapsto x_n = a_1 x_{n-1} + \cdots + a_k x_{n-k}
\]
### General Homogeneous Equation
Replace each instance of \( x_n \) with \( r^n \) to get the corresponding characteristic polynomial:
\[
r^n = \sum_{i=1}^{k} a_i r^{n-i}
\]
Solve for roots to determine solutions.
---
## Topic 2.6.2: Homogeneous Solutions
### Example
Given a homogeneous linear recurrence of degree \( k \):
\[
x_n = a_1 x_{n-1} + \cdots + a_k x_{n-k}
\]
Solve for the characteristic polynomial:
\[
r^k = a_1 r^{k-1} + \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004a8e3c-a278-4414-bbaf-77c8d726d7bd%2Fad8ef7ab-d8e3-4345-ba1c-c6e00f6a0ff3%2F4c31g6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Topic 2.6.0: Recurrence Relations
### Definition
Suppose \( k \in \mathbb{N} \) and \( f \) is a sequence function of the form:
\[
\begin{bmatrix}
x_0 \\
x_{n-1} \\
\vdots \\
x_{n-k+1}
\end{bmatrix}
\mapsto x_n = f(x_{n-1}, \ldots, x_{n-k})
\]
It is called a recurrence relation of order \( k \). If it has initial conditions such as:
\[
x_0 = a_0, \ldots, x_{k-1} = a_{k-1}
\]
Then the recurrence relation is of **order** \( k \).
### Examples include:
1. **Geometric Progression (GP)**: \( x_n = rx_{n-1} \)
2. **The Tower of Hanoi Sequence**: \( x_n = 2x_{n-1} + 1 \)
3. **The Fibonacci Sequence**: \( x_n = x_{n-1} + x_{n-2} \)
---
## Topic 2.6.1: Characteristic Equations
### Example
Suppose \( k \in \mathbb{N} \) and it's a linear recurrence relation of order \( k \):
\[
\begin{bmatrix}
x_0 \\
x_{n-1} \\
\vdots \\
x_{n-k+1}
\end{bmatrix}
\mapsto x_n = a_1 x_{n-1} + \cdots + a_k x_{n-k}
\]
### General Homogeneous Equation
Replace each instance of \( x_n \) with \( r^n \) to get the corresponding characteristic polynomial:
\[
r^n = \sum_{i=1}^{k} a_i r^{n-i}
\]
Solve for roots to determine solutions.
---
## Topic 2.6.2: Homogeneous Solutions
### Example
Given a homogeneous linear recurrence of degree \( k \):
\[
x_n = a_1 x_{n-1} + \cdots + a_k x_{n-k}
\]
Solve for the characteristic polynomial:
\[
r^k = a_1 r^{k-1} + \
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