Math 442 Assignment 3 - Spring 2015
Due Friday, February 13th
Directions: Use LATEX to typeset your solutions to the following. You will print your
solutions and hand these in on Friday, February 13th in class.
1. Consider a lake of fixed volume V cubic meters which is being polluted by some contaminant. Assume that the lake is well mixed so the contamination is uniform throughout.
Let x(t) denote the concentration (in grams per cubic meter) of contaminant at time
t ≥ 0 and let r > 0 denote the rate (in cubic meters per day) at which water flows in
(and out) of the lake. If pollution input is stopped, a model for lake purification is
r
dx
= − x,
dt
V
x(0) = x0 .
(1)
(a) If r is a constant, show that the solution of (1) is x(t) = x0 e−rt/V .
(b) Show that the time it takes for the level of pollution to be reduced to 5% of its
initial value is
V
V
t0.05 = − ln(0.05) = ln(20).
(2)
r
r
(c) Suppose there is an annual variation of the outflow rate r about its mean given
by r(t) = ρ[1 + sin(2πt)], where ρ is the mean outflow, t is the number of years
since pollution input ceased, and || < 1. (Here, ρ corresponds to r for = 0). In
this case, the lake purification model is given by
r(t)
dx
=−
x,
dt
V
x(0) = x0 .
(3)
The following steps will be used to show that an annual variation of r about its
mean would not significantly alter the value predicted for t0.05 in (2).
i. Show that the solution of (3) is
h ρ i
x(t) = x0 exp −
t+
(1 − cos(2πt)) .
V
2π
ii. Show that the time it takes for the level of pollution to be reduced to 5% of
its initial value satisfies
V
cos(2πt0.05 ) = t0.05 +
+ ln(0.05).
2π
2π
ρ
iii. Use the trigonometric identity cos(2θ) = 1 − 2 sin2 (θ) to show that
−
V
||
V
||
ln(0.05) −
≤ t0.05 ≤ − ln(0.05) + .
ρ
π
ρ
π
The last result shows that an annual variation of r about its mean would alter
the value of t0.05 in (2) by at most 1/π years (approximately 4 months).
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2. Consider the radioactive decay model
dx
= −λx,
dt
x(0) = x0 .
(a) Show that the half-life of a substance is given by t0.5 =
ln(2)
.
λ
(b) Polonium 210 has a half-life of 140 days. How long would it take a sample to
decay to 20% of its original amount?
3. Let x(t) denote the dry weight of a plant at time t ≥ 0 that feeds off a fixed amount
of a single substrate and let S(t) denote the weight of the remaining substrate at time
t ≥ 0. Assuming no material is lost when S is converted to x, a plant growth model is
dx
= k(xf − x)x,
dt
x(0) = x0
where k > 0 and xf > 0 denotes the value of x corresponding to S = 0.
(a) Show that the solution of this initial-value problem is
x(t) =
x0 xf
.
x0 + (xf − x0 )e−kxf t
(b) Compute lim x(t) and interpret your result.
t→∞
4. Two plants are growing independently off independent fixed supplies of substrate. Let
x(t) and y(t) denote the dry weights of these plants at time t ≥ 0. A plant growth
model is
dx
= kx(xf − x),
dt
dy
= ky(xf − y).
dt
(a) Write down the solution x(t) and y(t) of these equations.
(b) Suppose that k = 0.1, xf = 7, yf = 10, x(0) = 1, and y(0) = 3. Plot the solution
curves x(t) and y(t) on a single figure. Be sure to include axis labels and a legend.
(c) For k = 0.1, xf = 7, and yf = 10, plot solution curves in the xy-plane for several
different initial values x0 and y0 in the rectangle 0 < x < 13, 0 < y < 16.
5. Suppose that the lake in Problem 1 is being polluted at a rate of P tonnes (1000 kg)
per day. That is, at a rate of P × 106 grams per day.
(a) Show that a modified model for lake pollution is
dx
P × 106
r
=
− x,
dt
V
V
x(0) = x0 .
(b) Use an integrating factor to find a solution x(t) of this initial-value problem and
compute lim x(t).
t→∞
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6. Consider the Lotka-Volterra predator-prey model
dx
= (a1 − b1 y)x,
dt
dy
= (−a2 + b2 x)y.
dt
Suppose that a phase-plane diagram is drawn for general parameter values a1 , a2 , b1
and b2 . Use the relationship
dy
dy/dt
=
dx
dx/dt
and separation of variables to show that the solution curves belong to the family
a2 ln(x) − b2 x + a1 ln(y) − b1 y = ln(c),
where c is a constant.
7. Consider the Lotka-Volterra competition model
dx
= (a1 − b1 y)x,
dt
dy
= (a2 − b2 x)y.
dt
Suppose that a phase-plane diagram is drawn for general parameter values a1 , a2 , b1
and b2 . Show that the solution curves belong to the family
−a2 ln(x) + b2 x + a1 ln(y) − b1 y = ln(c),
where c is a constant.
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