/*! 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} Q11E Solve the nonlinear differential... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Q11E (page 425)

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxas dxfx=dtand integrate both sides.

10.dxdt=1+x2,x(0)=0

Short Answer

Expert verified

The solution isx(t)=tant.

Step by step solution

01

Explanation of the differential equation

Consider the equation as follows:

dxdt=1+x2

Now, separate the variables as follows.

role="math" localid="1659594666235" dxdt=1+x2dx1+x2=dt

Integrating on both sides as follows:

dx1+x2=dt鈭�dx1+x2=鈭�dttan-1x=t+C

Substituting the initial condition as follows:

tan-1x=t+Cx(t)=tan(t+C)x(0)=tan(0+C)0=0+C

Simplify further as follows:

C=0

02

Calculation of the solution

Now, substitute the value 0 for C in x(t)=tan(t+C) as follows.

x(t)=tan(t+C)x(t)=tan(t+0)x(t)=tant

Hence, the solution for the differential equation dxdt=1+x2is x(t)=tant.

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

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxas dxfx=dtand integrate both sides.

10.dxdt=1cos(x),x(0)=0

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=x+ydydt=2x-y|

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?

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxasdxfx=dtand integrate both sides.

6.dxdt=1x,x(0)=1

Solve the differential equationf'(t)鈭�5f(t)=0and find all real solutions of the differential equation.

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?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.