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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 3: Q1E (page 101)

The numberof supermarkets throughout the country that
are using a computerized checkout system is described by the
Initial-value problem
Use the phase portrait concept of Section to predict
how many supermarkets are expected to adopt the new procedure over a long period. By hand, sketch a
solution curve of the given initial-value problem.
(b) Solve the initial-value problem and then use a graphing
utility to verify the solution curve in part (a). How many
companies are expected to adopt the new technology when
?

Short Answer

Expert verified

Answer

(a)

(b)

Step by step solution

01

Find the equilibrium solutions by solving

02

Find Equilibrium solutions

Therefore equilibrium solutions are and

03

The solution curve is shown below;

That means supermarkets are expected to adopt the new procedure over a long period.

04

This is a logistic differential equation.

We know that the solution of a logistic differential equation is:

We haveand

05

The graph is shown below

06

Step 6:

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Most popular questions from this chapter

A classical problem in the calculus of variations is to find the shape of a curve such that a bead, under the influence of gravity, will slide from point $A (0, 0)$ to point $B (x_{1}, y_{1})$ in the least time. See Figure 3.R.3. It can be shown that a nonlinear differential for the shape $y (x)$ of the path is $y [1 + {(y')}^{2}] = k$ , where $k$ is a constant. First solve for k $d x$ in terms of $y$ and $d y$ , and then use the substitution $y = k s i n^{2} θ$ to obtain a parametric form of the solution. The curve turns out to be a cycloid.

FIGURE3.R.3 Sliding bead in Problem 12

Repeat problem 6 in case \({\bf{a = 5,b = 1}}\) and \({\bf{h = 7}}\).

Evaporating Raindrop As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is

negligible, then a model for the velocity $v (t)$ of the raindrop is

$\frac{d v}{d t} + \frac{3 (\frac{k}{ρ})}{(\frac{k}{ρ}) t + r_{0}} v = g$

Here $ρ$ is the density of water, $r_{0}$ is the radius of the raindrop at $t = 0, k < 0$ is the constant of proportionality, and the downward direction is taken to be the positive direction.

(a) Solve for v(t) if the raindrop falls from rest.

(b) Reread Problem 36 of Exercises 1.3 and then show that the radius of the raindrop at time t is $r (t) = (\frac{k}{ρ}) t + r_{0}$

(c) If $r_{0} = 0.01 f t$ and $r = 0.007 f t 10 s e c o n d s$ after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely.

Question: When all the curves in a family \(G\left( {x,y,{c_1}} \right) = 0\)intersect orthogonally all the curves in another family\(H\left( {x,y,{c_2}} \right) = 0\), the families are said to be orthogonal trajectories of each other. See Figure 3.R.5. If \(dy/dx = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(dy/dx = - 1/f(x,y)\). In Problems\(\;{\bf{15}} - {\bf{18}}\)find the differential equation of the given family by computing\(dy/dx\)and eliminating \({c_1}\)from this equation. Then find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

\(y = {c_1}{e^x}\)

Rocket Motion Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. In Problem 22 of Exercises 1.3 you were asked to use Newton’s second law of motion in the form given in (17) of that exercise set to show that a mathematical model for the velocity v(t) of the rocket is given by

$\frac{d v}{d t} + \frac{k - v}{m_{0} - λ t} v = - g + \frac{R}{m_{0} - λ t}$ ,

where k is the air resistance constant of proportionality, is the constant rate at which fuel is consumed, R is the thrust of the rocket, $m (t) = m_{0} - λ t$ , $m_{0}$ is the total mass of the rocket at $t = 0$ , and g is the acceleration due to gravity. (a) Find the velocity $v (t)$ of the rocket if $m_{0} = 200 kg, R = 2000 N, λ = 1 kg / s, g = 9.8 m / s^{2}, k = 3 k g / s$ and $v (0) = 0$ .

(b) Use $d s / d t = v$ and the result in part (a) to nd the height s(t) of the rocket at time t .

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