Math 308 - Differential Equations Fall 2002
Homework Assignment 2
Due Friday, September 20.
1. On the last page of this assignment are five differential equations for u(t), and five slope fields. Match
the equation with the slope field.
2. In each of the slope fields in Problem 1, sketch the function u(t) for which u(0) = 1.
3. Consider an object falling through a viscous medium. According to Newton’s second law of motion,
the motion of an object is governed by the equation F=ma, where Fis the total force acting on the
object, mis the mass of the object, and ais the acceleration of the object. If we let y(t) be the height
of the object, then v(t) = dy
dt is the velocity, and a(t) = dv
dt =d2y
dt2is the acceleration. Note that with
this choice of y, a positive velocity (v > 0) means the object is moving up.
The forces acting on the object are gravity (a constant force given by mg, where gis the acceleration
due to gravity), and friction. The frictional force depends on how fast the object is moving, and is
always directed in the opposite direction of the motion. Let Fd(v) be the force due to friction (the d
subscript refers to drag).
(a) Write down the differential equation satisfied by v, the velocity of the object. Assume that g > 0
(so you must be sure to put the correct sign in the equation), and that when v > 0, Fd(v)<0.
If the object is released from height y0when t= 0, what is the initial condition for the differential
equation?
(b) Consider no friction, so Fd(v) = 0. Solve the initial value problem for v(t). What happens to v
at tincreases? That is, what is the long-term behavior of v?
(c) Suppose the friction force is proportional to the velocity. That is, let Fd(v) = −Cdv, where
Cd>0 is a constant (the drag coefficient). Solve the initial value problem. What happens to v
as tincreases?
(d) Now suppose that the friction force is proportional to the square of the velocity: let Fd(v) =
−Cdv|v|. Solve the new problem, and describe the long-term behavior.
Note that v|v|=(v2v≥0
−v2v≤0. I’ve written it this way to ensure that the friction force always
opposes the velocity.
Also, observe that for any non-zero constant a,
1
v2−a2=1
(v+a)(v−a)=−1
2a(v+a)+1
2a(v−a).
This is an example of a partial fraction expansion. It can be useful when you try to integrate the
separated equation.
(e) If yis measured in meters, tin seconds, and mass in kilograms, what must the units of Cdbe in
the previous two questions? (Note: the units are not the same in the two cases.)
4. The logistic equation
dP
dt =kµ1−P
N¶P
is a first order autonomous differential equation, so it is separable. Find the general solution to this
differential equation.
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