What's the general solution to the following second order differential equation if A and B are constants? Why? y" +9y=0
What's the general solution to the following second order differential equation if A and B are constants? Why? y" +9y=0
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
![Title: Solving Second Order Differential Equations
## Problem Statement
What's the general solution to the following second order differential equation if A and B are constants? Why?
\[
y'' + 9y = 0
\]
## Solution Explanation
This is a homogeneous linear differential equation with constant coefficients. The general approach to solve such equations is to assume a solution of the form:
\[
y = e^{rt}
\]
Substituting into the differential equation, we obtain the characteristic equation:
\[
r^2 + 9 = 0
\]
Solving this, we find the roots:
\[
r = \pm 3i
\]
These are complex roots, indicating that the general solution will be a combination of sine and cosine functions:
\[
y(t) = A \cos(3t) + B \sin(3t)
\]
Here, \( A \) and \( B \) are arbitrary constants determined by initial conditions. The solution reflects harmonic motion, typical for systems like simple harmonic oscillators.
This form of the solution utilizes Euler's formula, which connects complex exponentials to trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67c02a1a-91b6-4dde-807f-4ee7d9512ede%2F4ec5f1f2-8bf6-44d0-9bf7-d6614f8ea3f4%2Ffh9svpr_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Solving Second Order Differential Equations
## Problem Statement
What's the general solution to the following second order differential equation if A and B are constants? Why?
\[
y'' + 9y = 0
\]
## Solution Explanation
This is a homogeneous linear differential equation with constant coefficients. The general approach to solve such equations is to assume a solution of the form:
\[
y = e^{rt}
\]
Substituting into the differential equation, we obtain the characteristic equation:
\[
r^2 + 9 = 0
\]
Solving this, we find the roots:
\[
r = \pm 3i
\]
These are complex roots, indicating that the general solution will be a combination of sine and cosine functions:
\[
y(t) = A \cos(3t) + B \sin(3t)
\]
Here, \( A \) and \( B \) are arbitrary constants determined by initial conditions. The solution reflects harmonic motion, typical for systems like simple harmonic oscillators.
This form of the solution utilizes Euler's formula, which connects complex exponentials to trigonometric functions.
Transcribed Image Text:What's the general solution to the following second order differential equation if A and B are constants? Why?
\( y'' + 9y = 0 \)
Expert Solution
Step 1: Given equation
Consider given differential equation,
y"+9y = 0
Associated auxiliary equation is
m ^2 + 9 = 0
ie m ^2 = -9
m = 3i , -3i
Therefore auxillary equation has complex roots.
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