Title: Solving Second Order Differential Equations## Problem StatementWhat's the general solution to the following second order differential equation if A and B are constants? Why?\[y'' + 9y = 0\]## Solution ExplanationThis 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. What's the general solution to the following second order differential equation if A and B are constants? Why?\( y'' + 9y = 0 \)

Answered: What's the general solution to the…

Math

Advanced Math

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

See similar textbooks

Related questions

Q: 2. ху' %3D2хе* -у + 6x2

Q: An IVP ordinary differential equation is called well-posed if a big change in the data of the…

A: We know that an IVP ordinary differential equation is called well posed if a It has a solution b…

Q: Obtain the differential equation of the following

Q: Solve the following differential equation. i. у" +11у' + 24у %3D0 i. у" — 4у' +9у %3D0 ii. у" — 4y'…

Q: -2r y=Ae²+ Be -2x-5e- y=Ae-2¹ cos(x) + Be-2¹ sin(x)

Q: Find a differential equation whose general solution is y = c₁e²t + C₂e-2.

A: Note: Your general solution should be: .The differential equation for this general solution needs to…

Q: A tank initially contains 50 gallons of brine, with 30 pounds of salt in solution. Water runs into…

A: This question is related to differrential equations, We will solve it using given information.

Q: A red wine at room temperature 70°F is chilled in ice (32°F). If it takes 15 minutes for the wine to…

Q: ず-8g+y ガ-8g+y=(x+8x) ex パ+82

Q: Solve the differential equation; yy + 25x = 0 %3D

Q: (y – c,)² + (x – c2)² = 1 %3D -

Q: What is the equation of the line that passes through (2,4) and has a slope or −1 in point-slope…

A: .

Q: А. у"+8у'+41у%3D0 B. 2y"-y'-y= 0 %3D

Q: Find a differential equation whose general solution is y = cet + c₂e-4² = 0

Q: The differential equation d y 3x4 2- (3 In x)y = - dx2 is a Bernoulli's equation. Select one: O True…

Q: The change of Turkey's population (N) over time (t) is proportional to the current population. In…

A: Note: It is assumed that the population of 2021 is being asked this question. The general solution…

Q: у" — Зу" + 4у — 6+ 80сos(2x)

Q: 2(2x? + y?)dx – xydy = 0 y = 1; x = 1

Q: The differential equation that represent the following system 1-1 X, is 2 3 Select one: a. y"-4y-5y…

A: The given system is

Q: Let ao, a1, a2, ... be the sequence defined recursively by an = 7(an-1)² if n ≥ 1 ao = 1, Use the…

A: Principle of mathematical induction states that a statement P(n) is true for all natural numbers n…

Q: Find the Maclaurin series of O Σ(x)3n n=0 3 1+x 3 Σ(-1)*(x)" n=0 3 Σω n=0 Write your answer in sigma…

Q: 7. у"" + у" — 7у" — у' + бу %3D 0 - -

A: On solving this differential equation we will get

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.

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.

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

B- Solve the following differential equation у" — 7у' + 3 %3D о
Get a first-order differential equation to solve current in a LR circuit analysis. The voltage of 20sin(t) is applied. The inductance is0.8 henry resistor is 100 and the lo current is 0! As I journey into DQ I would like an in-depth solution. Show all work and rules, equations. I want to have the foundation.
What is the order of the differential equation y"-3y + 6y-2y=0? Enter your answer as a numerical answer. For example, if the equation is seventh-order, then enter 7 as your answer.
State the order and degree of each differential Equation
If y=e" is a solution of the differential equation y" - By + ky0, and A=2 and B= 6, then the value of "K" is
What do you mean by an implicit solution of a differential equation?