/*! 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} Q2E Use the concept of a continuous ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Q2E (page 425)

Use the concept of a continuous dynamical system.Solve the differential equation dxdt=鈭�kx. Solvethe system dx鈫�dt=Ax鈫�when Ais diagonalizable overR,and sketch the phase portrait for 2脳2 matricesA.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.

2. dxdt=-0.71xwith x(0)=-e.

Short Answer

Expert verified

The solution isy=-e1-0.71t

Step by step solution

01

Step 1:Definition of the differential equation

Consider the differential equation dydx=kxwith initial value x0(k is an arbitrary constant). The solution is .x(t)=x0ekt

The solution of the linear differential equationdydx=kxandy(0)=C is .y=Cekx

02

Calculation of the solution

Given the differential equation dxdt=-0.71xwith the initial conditionx(0)=-e

Substitute in the solutiony=Cekx as follows.

y=Cekxy=-ee-0.71ty=-e1-0.71t

Hence, the solution for the differential equationdxdt=-0.71x isy=-e1-0.71t

03

Graphical representation of the solution

The graph of the equation y=-e1-0.71tis sketched below as follows.

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!

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

Consider an n脳n matrix A with m distinct eigenvalues 位1,位2,鈥�,位m.

(a) Show that the initial value problemdx鈫�dt=Ax鈫� withrole="math" localid="1660807946554" x鈫�(0)=x鈫�0 has a unique solutionrole="math" localid="1660807989045" x鈫�(t)

(b) Show that the zero state is a stable equilibrium solution of the systemdx鈫�dt=Ax鈫� if and only if the real part of all the位i is negative.Hint: Exercise 47 and Exercise 8.1.45 are helpful.

Question: Consider the system dx鈫�dt=[01-10]Ax鈫� whereA=[0100]. Sketch a direction field for Base on your sketch, describe the trajectories geometrically. Can you find the solution analytically?

Question: Consider the interaction of two species in a habitat. We are told that the change of the populationsx鈫�(t)andy鈫�(t) can be moderated by the equation

|dxdt=1.4x-1.2ydxdt=1.8x-1.4y|

where timeis a measured in years.

  1. What kind of intersection do we observe (symbiosis, competition, or predator-prey)?
  2. Sketch the phase portrait for the system. From the nature of the problem, we are interested only in the first quadrant.
  3. What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Use theorem 9.3.13 to solve the initial value problem

dx鈫�dt=(1201)x鈫�withx鈫�(0)=[1-1] .

Hint: Find first x2(t) and then x1(t).

Consider a systemdx鈫�dt=Ax鈫�where A is a symmetric matrix. When is the zero state a stable equilibrium solution? Give your answer in terms of the definiteness of the matrix A.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.