Homework 5
For the following two problems, seperate variables and use partial fractions
to solve the givne initial value problems.
1.
2.
3.
x0 = x x2 ; x(0) = 2
x0 = 9 4x2 ; x(0) = 0
The time rate of change of a rabbit population P is proportional to the
square root of P . At time t = 0 (months), the population numbers 100
rabbits and is increasing at a rate of 20 rabbits per month. How many
rabbits will there be one year later?
4. Consider a population P (t) satisfying the extinction-explosion equation
dP=dt = aP 2 bP , where B = aP 2 is the time rate at which births
occur and D = bP is the time rate at which deaths occur. If the initial
population is P (0) = P0 and B0 births per month and D0 deaths per
month are occuring at time t = 0, show that the threshold population is
M = D0 P0 =B0 .
5. As the salt KNO3 dissolves in methanol, the number x(t) of grams of
the salt in a solution after t seconds satises the dierential equation
dx=dt = 0:8x 0:004x2.
(a) What is the maximum of the salt that will ever dissolve in the
methanol?
(b) If x = 50 when t = 0, how long will it take for an additional 50 grams
of salt to dissolve?
6. 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) = v0 e kt
x(t) = x0 + vk0 (1 e kt ):
(b) Conclude that the body travels only a nite distance and nd that
distance.
7. Suppose that
a car starts from rest, its engine providing an acceleration
of 10 ft/s2 , while air resistance provides 0.1 ft/s2 of deceleration for each
foot per second of the car's velocity.
(a) Find the car's maximum possible velocity.
(b) Find how long it takes the car to attain ninety percent of its limiting
velocity, and how far it travels while doing so.
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8. A woman bails out of an airplane at an altitude of ten thousand feet, falls
freely for twenty seconds , then opens her parachute. Assume linear air
resistance v ft/s2 , taking = 0:15 without the parachute and = 1:5
with the parachute. Suggestion: First determine her height above the
ground and velocity when the parachute opens.
9. The mass of the sum is 329,320 times that of the earth (Mearth = 5:975 1024 kg) and its radius is 109 times the radius of the earth (Rearth =
6:378106 m). Assume that the gravitational constant G = 6:672610 11
N (m=kg)2 .
(a) To what radius (in meters) would the earth have to be compressed in
order for it to become a black hole ? That is, for the escape velocity
to equal the velocity c = 3 108 m/s of light?
(b) Repeat part (a) with the sun in place of the earth.
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