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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 271
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.

dxdt=y-1,dydt=ex+y

The solution is ex+ye-y=c .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find phase plane equation

Here the system is;

dxdt=y-1dydt=ex+y

And the phase plane equation is;

dydx=ex+yy-1

Step 2: Solve the equation

Here the equation is.

dydx=ex+yy-1

Solve by variable separating,

∫(y-1)eydy=∫exdx-ye-y+c=exex+ye-y=c

This is the required result.

Most popular questions for Math Textbooks

Using the software, sketch the direction field in the phase-plane for the system dxdt=y,dydt=-x-x3. From the sketch, conjecture whether all solutions of this system are bounded. Solve the related phase plane differential equation and confirm your conjecture.

A house, for cooling purposes, consists of two zones: the attic area zone A and the living area zone B (see Figure 5.4). The living area is cooled by a 2 – ton air conditioning unit that removes 24,000 Btu/hr. The heat capacity of zone B is 12F∘ per thousand Btu. The time constant for heat transfer between zone A and the outside is 2 hr, between zone B and the outside is 4 hr, and between the two zones is 4 hr. If the outside temperature stays at 100°F, how warm does it eventually get in the attic zone A? (Heating and cooling buildings was treated in Section 3.3 on page 102.)

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dxdt=3ydydt=2x

Logistic Model. In Section 3.2 we discussed the logistic equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{2}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)and its use in modeling population growth. A more general model might involve the equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{r}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)where r>1. To see the effect of changing the parameter r in (25), take \({{\bf{p}}_{\bf{1}}}\)= 3, A = 1, and \({{\bf{p}}_{\bf{o}}}\)= 1. Then use a numerical scheme such as Runge–Kutta with h = 0.25 to approximate the solution to (25) on the interval\(0 \le {\bf{t}} \le 5\) for r = 1.5, 2, and 3What is the limiting population in each case? For r >1, determine a general formula for the limiting population.

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

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