/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 For each of the differential equ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 39

For each of the differential equations in find the values of \(b\) that make the general solution: (a) overdamped, (b) underdamped, (c) critically damped. $$s^{\prime \prime}+b s^{\prime}+5 s=0$$.

Short Answer

Expert verified
(a) Overdamped: \(b > \sqrt{20}\) or \(b < -\sqrt{20}\). (b) Underdamped: \(-\sqrt{20} < b < \sqrt{20}\). (c) Critically damped: \(b = \sqrt{20}\) or \(b = -\sqrt{20}\).

Step by step solution

01

Recognize the Standard Form of the Equation

The given differential equation is a second-order linear homogeneous differential equation of the form:\[ s'' + bs' + 5s = 0 \]Recognize it as a **second-order linear differential equation with constant coefficients**.
02

Write the Characteristic Equation

For the equation \( s'' + bs' + 5s = 0 \), write the characteristic equation:\[ r^2 + br + 5 = 0 \]This characteristic equation helps in determining the type of damping (overdamped, underdamped, or critically damped).
03

Use the Discriminant to Determine Damping Type

The type of damping can be determined by the discriminant \(D\) of the characteristic equation:\[ D = b^2 - 4ac \]In our case, \(a = 1\), \(b\) is the coefficient of \(s'\), and \(c = 5\). Therefore, the discriminant is:\[ D = b^2 - 20 \].
04

Find Overdamped Condition

The system is overdamped when \(D > 0\), meaning:\[ b^2 - 20 > 0 \]Solving for \(b\), we find:\[ b^2 > 20 \]Thus, \(b > \sqrt{20}\) or \(b < -\sqrt{20}\).
05

Find Underdamped Condition

The system is underdamped when \(D < 0\), meaning:\[ b^2 - 20 < 0 \]Solving for \(b\), we find:\[ b^2 < 20 \]Thus, \(-\sqrt{20} < b < \sqrt{20}\).
06

Find Critically Damped Condition

The system is critically damped when \(D = 0\), meaning:\[ b^2 - 20 = 0 \]Solving for \(b\), we find:\[ b^2 = 20 \]Thus, \(b = \sqrt{20}\) or \(b = -\sqrt{20}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Second-Order Linear Differential Equation
A second-order linear differential equation is an equation that involves the second derivative of a function. In its simplest form, it looks like this:
  • \( ay'' + by' + cy = 0 \)
This type of equation is common in physics and engineering, especially when describing systems such as springs or circuits.
In such equations, \( a \), \( b \), and \( c \) are constants, and \( y'' \), \( y' \), and \( y \) are the second derivative, first derivative, and the original function, respectively.
The problem we are looking at is in this standard form but uses different variables:
  • \( s'' + bs' + 5s = 0 \)
Understanding this form is crucial for solving second-order linear differential equations.
Characteristic Equation
The characteristic equation comes into play when trying to solve second-order linear differential equations. This equation is derived from the original differential equation by assuming solutions of the form \( e^{rt} \), where \( r \) is a number to be determined.
For our equation, \( s'' + bs' + 5s = 0 \), the corresponding characteristic equation is:
  • \( r^2 + br + 5 = 0 \)
This quadratic equation helps to determine the type of roots the solution will have. Knowing the roots helps in understanding the nature of the solution, particularly how the system behaves over time.
The roots can be real and distinct, real and repeated, or complex, and each case corresponds to a different type of damping.
Damping Types
Damping is an essential concept when examining differential equations that model real-world systems. By analyzing the roots of the characteristic equation, we can determine the system's damping type:
  • **Overdamped:** This occurs when the characteristic equation has two distinct real roots (\( D > 0 \)). In this state, the system returns to equilibrium without oscillating, but it takes longer.
  • **Underdamped:** This happens when the roots are complex (\( D < 0 \)). When a system is underdamped, it oscillates while gradually returning to equilibrium, like a slowly settling spring.
  • **Critically damped:** This is the state where the system returns to equilibrium as quickly as possible without oscillating. Here, the characteristic equation has repeated real roots (\( D = 0 \)).
Understanding these types helps in designing systems for stability and efficiency.
Discriminant
The discriminant, \( D \), is a vital component in determining the solutions to quadratic equations and, consequently, the damping type of a system.
For the characteristic equation \( r^2 + br + 5 = 0 \), the discriminant is calculated as:
  • \( D = b^2 - 4ac \)
In our specific case, with \( a = 1 \), \( b = b \), and \( c = 5 \), the discriminant becomes:
  • \( D = b^2 - 20 \)
This expression, \( b^2 - 20 \), determines whether we have overdamping, underdamping, or critical damping.
  • If \( D > 0 \), we have overdamping.
  • If \( D < 0 \), we have underdamping.
  • If \( D = 0 \), we reach critical damping.
Knowing the discriminant gives a straightforward method to categorize the system's behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each of the differential equations find the values of \(c\) that make the general solution: (a) overdamped, (b) underdamped,(c) critically damped. $$s^{\prime \prime}+2 \sqrt{2} s^{\prime}+c s=0$$

Let \(L,\) a constant, be the number of people who would like to see a newly released movie, and let \(N(t)\) be the number of people who have seen it during the first \(t\) days since its release. The rate that people first go see the movie, \(d N / d t\) (in people/day), is proportional to the number of people who would like to see it but haven't yet. Write and solve a differential equation describing \(d N / d t\) where \(t\) is the number of days since the movie's release. Your solution will involve \(L\) and a constant of proportionality, \(k.\)

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(x)\) is even, then \(f(x)\) is odd.

Analyze the phase plane of the differential equations for \(x, y \geq 0 .\) Show the nullclines and equilibrium points, and sketch the direction of the trajectories in each region. $$\begin{aligned}&\frac{d x}{d t}=x\left(1-x-\frac{y}{3}\right)\\\&\frac{d y}{d t}=y\left(1-y-\frac{x}{2}\right)\end{aligned}$$

Give an example of: A system of differential equations for the profits of two companies if each would thrive on its own but the two companies compete for business. Let \(x\) and \(y\) represent the profits of the two companies.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.