Math 2280 - Assignment 3

Dylan Zwick

Spring 2013

Section 2.1 - 1, 8, 11, 16, 29

Section 2.2 - 1, 10, 21, 23, 24

Section 2.3 - 1, 2, 4, 10, 24

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Section 2.1 - Population Models

2.1.1 Separate variables and use partial fractions to solve the initial valueproblem:

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2.1.8 Separate variables and use partial fractions to solve the initial valueproblem:

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2.1.11 Suppose that when a certain lake is stocked with fish, the birth anddeath rates 3 and 5 are both inversely proportional to

(a) Show that

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(b) If P0 = 100 and after 6 months there are 169 fish in the lake, howmany will there be after 1 year?

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2.1.16 Consider a rabbit population P(t) satisfying the logistic equationdP/dt = aP — bP2. If the initial population is 120 rabbits and thereare 8 births per month and 6 deaths per month occuring at time t 0,how many months does it take for P(t) to reach 95% of the limitingpopulation M?

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2.1.29 During the period from 1790 to 1930 the U.S. population P(t) (t inyears) grew from 3.9 million to 123.2 million. Throughout this period, P(t) remained close to the solution of the initial value problem

= 0.03135P — 0.0001489P2, P(0) 3.9.

(a) What 1930 population does this logistic equation predict?

(b) What limiting population does it predict?

(c) Has this logistic equation continued since 1930 to accurately modelthe U.S. population?

[This problem is based on the computation by Verhuist, who in 1845used the 1790-1840 U.S. population data to predict accurately theU.S. population through the year 1930 (long after his own death, ofcourse).]

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Section 2.2 - Equilibrium Solutions and Stability

2.2.1 - Find the critical points of the autonomous equation

dx—x-4.dt

Then analyze the sign of the equation to determine whether eachcritical point is stable or unstable, and construct the correspondingphase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either theexact solution or a computer-generated slope field to sketch typicalsolution curves for the given differential equation, and verify visually the stability of each critical point.

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2.2.10 Find the critical points of the autonomous equation

= 7x — — 10.dt

Then analyze the sign of the equation to determine whether eachcritical point is stable or unstable, and construct the correspondingphase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either theexact solution or a computer-generated slope field to sketch typicalsolution curves for the given differential equation, and verify visually the stability of each critical point.

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2.2.21 Consider the differential equation dx/dt = kx —

(a) If k 0, show that the only critical value c = 0 of x is stable.

(b) If k > 0, show that the critical point c = 0 is now unstable, butthat the critical points c = are stable. Thus the qualitativenature of the solutions changes at k = 0 as the parameter k increases, and so k 0 is a bifurcation point for the differentialequation with parameter Ic.

The plot of all points of the form (k, c) where c is a critical point ofthe equation x’ = kx — x3 is the “pitchform diagram” show in figure2.2.13 of the textbook.

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2.2.23 Suppose that the logistic equation cLr/dt = kx(M—x) models a population x(t) of fish in a lake after t months during which no fishingoccurs. Now suppose that, because of fishing, fish are removed fromthe lake at a rate of hx fish per month (with h a positive constant).Thus fish are “harvested” at a rate proportional to the existing fishpopulation, rather than at the constant rate of Example 4 from thetextbook.

(a) If 0 < h < kM, show that the population is still logistic. What isthe new limiting population?

(b) If h> kM. show that x(t) — 0 as t — oc, so the lake is eventuallyfished out.

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2.2.24 Separate variables in the logistic harvesting equation

dx/dt = k(N — x)(x — H)

and then use partial fractions to derive the solution given in equation15 of the textbook (also appearing in the lecture notes).

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Section 2.3 - Acceleration-Velocity Models

2.3.1 The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. If the machinecan accelerate from rest to 100 km/h in lOs, how long will it take forthe car to accelerate from rest to 200 km/h?

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2.3.2 Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = —kv.

(a) Show that its velocity and position at time t are given by

v(t) =

and

(t) = XO+ () (1_ e).

(b) Conclude that the body travels only a finite distance, and findthat distance.

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2.3.4 Consider a body that moves horizontally through a medium whoseresistance is proportional to the square of the velocity v, so that

dv/dt = —kv2.

Show that

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and that

x(t) = x0 + in (1 + uokt).

Note that, in contrast with the result of Problem 2, x(t) —* oc ast —* cxi Which offers less resistance when the body is moving fairlyslowly - the medium in this problem or the one in Problem 2? Doesyour answer seem to be consistent with the observed behaviors of

as t —÷ oc?

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2.3.10 A woman bails out of an airplane at an altitude of 10,000 ft, fallsfreely for 20s, then opens her parachute. How long will it take herto reach the ground? Assume linear air resistance pv ft/s2. takingp = .15 without the parachute and p = 1.5 with the parachute. (Suggestion: First determine her height above the ground and velocitywhen the parachute opens.)

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2.3.24 The mass of the sun is 329,320 times that of the earth and its radiusis 109 times the radius of the earth.

(a) To what radius (in meters) would the earth have to be compressedin order for it to become a black hole - the escape velocity fromits surface equal to the velocity c = 3 x 108m/s of light?

(b) Repeat part (a) with the sun in place of the earth.

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