Math 308 - Differential Equations Fall 2002
Homework Assignment 3
Due Friday, September 27.
1. Consider the initial value problem
dy
dt =y
t
(a) Determine the region in the (t, y) plane where the Existence and Uniqueness Theorems
apply.
(b) Pick 16 or so points in the (t, y) plane, and plot a slope field. Include some negative
values of tand y. Based on the slope field, can you guess the solutions?
(c) Solve the differential equation (it is separable). Sketch the solutions if y(1) = 0, y(1) = 1,
or y(1) = 2.
(d) What is y(0) for each of the solutions that you found in (c)? Does this contradict the
Uniqueness Theorem?
2. Find all the solutions to the initial value problem
dy
dt = 5y4
5, y(0) = 0.
3. For each of the following differential equations: sketch the phase line; find the equilibrium
points and classify them as sinks, sources, or nodes; in one graph, sketch the equilibrium
solutions along with several representative solution curves versus t. (Note: You do not have
to solve the differential equations analytically.)
(a) dy
dt =y2−6y−16
(b) dy
dt =y2+ 2y+ 10
(c) dy
dt = (y+ 2)(y−1)2
(d) dy
dt =−2y+ sin y
4. In Problem 3 of Homework 2, you solved the problem of an object falling that is also acted on
by friction. The differential equation was
dv
dt =−g+Fd(v)
m,
where gis the gravitational acceleration (g > 0), mis the mass of the object, and Fd(v)
is the friction force for velocity v. The three forms of friction considered were Fd(v) = 0,
Fd(v) = −Cdvand Fd(v) = −Cdv|v|.
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