Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #11
(32 points)
Worksheet #11 is due Thursday, December 10th . You are encouraged to work with other people
to solve these problems, but you have to write the solutions down individually. Please turn in these
sheets only. You can write at the bottom of each page and/or on the back. Show all the steps
and explain your reasoning when necessary: correct solutions with no explanation do not count as
complete. Partial credit will be assigned to incomplete answers.
Problem 1. (12 points) Match each of the following differential equations with the corresponding
direction field below. Do not attempt to solve the equations, just use their distinctive features instead
(and explain your reasoning). Also, for each differential equation draw a sketch of the solution with
y(0) = 0, using the information given by the direction field.
(a) y 0 = 2 − y
(b) y 0 = x(2 − y)
(c) y 0 = sin x sin y
Problem 2. (20 points) A man is walking his dog along the x axis of a Cartesian plane, using a leash
that is 1 unit long. When the dog reaches the origin, and the man is on the point (1, 0), a cat comes
out of nowhere and starts running along the y axis. The dog immediately runs after it, dragging its
owner along. Since the leash is always stretched during this motion, the resulting path of the man
is given by the purple curve that you see in the picture below, which has the property to be always
tangent to the segment (the leash) that joins the man to the dog.
We want to find the equation for that curve (whose name is tractrix by the way), and we will do
it using differential equations.
(a) Say the man is on the point (x, y) along the curve. Recalling that the leash is 1 unit long, find
the slope of the segment M D in terms of x and y. [Warning: pay attention to the sign]
(b) Now say y = y(x) is the equation of the curve; imposing that the segment whose slope you found
in item (a) is tangent to the curve gives us a differential equation. Write down the differential
equation and solve it for the initial condition y(1) = 0.
[Hint: this is a separable equation. You may use the fact that
Z
cos u + 1
du
= − ln
+ C.]
sin u
sin u