/*! 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} Q. 46 Find and graph the vector functi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Q. 46 (page 860)

Find and graph the vector function determined r(t)=鉄�x(t),y(t)鉄�by the differential equation

x鈥�(t)=x,y鈥�(t)=x2,x(0)=1,y(0)=2. (HINT: Start by solving the initial-value problem x鈥�(t)=x,x(0)=1.)

Short Answer

Expert verified

Ans: Vector function is r(t)=<et,te2t+2>

Graph is:

Step by step solution

01

Step 1. Given information.

given,

x鈥�(t)=x,y鈥�(t)=x2,x(0)=1,y(0)=2

02

Step 2. The objective of this problem is to find and graph the vector r(t)=⟨x(t),y(t)⟩ determined by the differential equations

x鈥�(t)=x鈥�鈥�(1)y鈥�(t)=x2鈥�鈥�(2)x(0)=1,y(0)=2

solve equation 1

x'(t)=x

dxdt=x

Separate the variables

dxx=dt

Integrate both sides

鈭�dxx=鈭�dt+c1ln鈦�x=t+c1x=et+c1x=cet

Therefore,

x=et.....(3)

03

Step 3. Now solve 

y鈥�(t)=x2dydt=x2dy=x2dt

Integrate

鈭�dy=鈭�x2dt+c2y=x2t+c2Att=0,y=22=c2

Therefore,

y=x2t+2....(4)

Use x=etfrom (3) into (4)

y=te2t+2

Thus, the vector of the function is

r(t)=鉄�et,te2t+2鉄�

04

Step 4. Draw the graph

Graph is:

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

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(t,t2,t3)fort鈭�[0,2]

Let Cbe the graph of a vector-valued function r. The plane determined by the vectors Nt0,Bt0and containing the point rt0 is called the normal plane forC atrt0. Find the equation of the normal plane to the helix determined byr(t)=<cost,sint,t>fort=蟺.

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domainlocalid="1649578696830" t鈮�0,r0=0,3,r1=-2,1,limt鈫�鈭�x(t)=-5,andlimt鈫�鈭�y(t)=-鈭�.

Lett=f(蟿)beadifferentiablereal-valuedfunctionof蟿,andletr(t)beadifferentiablevectorfunctionwitheithertwoorthreecomponentssuchthatf(蟿)isinthedomainofrforeveryvalueof蟿onsomeintervalI.Finddrf蟿d蟿.

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.

(1,t,t2)脳(1,t2,t3)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.