6) Find the general solhution to the fourth order differential equation is equation "whose"' characteristic
6) Find the general solhution to the fourth order differential equation is equation "whose"' characteristic
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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Question
![**Problem Statement:**
Find the general solution to the fourth-order differential equation whose characteristic equation is:
\[
(r + 2)(r^2 + 4)r = 0
\]
**Explanation:**
The characteristic equation is expressed as a product of factors, which represents a polynomial in terms of \( r \). Solving this characteristic equation will help determine the roots, which are used to construct the general solution of the differential equation. Here, the factors can be broken down as follows:
1. \( r = 0 \)
2. \( r + 2 = 0 \) leading to \( r = -2 \)
3. \( r^2 + 4 = 0 \) leading to \( r^2 = -4 \) or \( r = \pm 2i \)
Each root contributes to the general solution of the differential equation based on whether the roots are real, repeated, or complex.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cc4fb4a-23f5-4110-9e16-75ef5a899c17%2F91fc40f5-17c4-4319-9dab-934696dc8660%2F9axxcfu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the general solution to the fourth-order differential equation whose characteristic equation is:
\[
(r + 2)(r^2 + 4)r = 0
\]
**Explanation:**
The characteristic equation is expressed as a product of factors, which represents a polynomial in terms of \( r \). Solving this characteristic equation will help determine the roots, which are used to construct the general solution of the differential equation. Here, the factors can be broken down as follows:
1. \( r = 0 \)
2. \( r + 2 = 0 \) leading to \( r = -2 \)
3. \( r^2 + 4 = 0 \) leading to \( r^2 = -4 \) or \( r = \pm 2i \)
Each root contributes to the general solution of the differential equation based on whether the roots are real, repeated, or complex.
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