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Unformatted text preview: Name:
TA: PID:
Sec. No: Sec. Time: Math 20D.
Midterm Exam 1
October 24, 2008 Turn oﬀ and put away your cell phone.
No calculators or any other electronic devices are allowed during this exam.
You may use one page of notes, but no books or other assistance during this exam.
Read each question carefully, and answer each question completely.
Show all of your work; no credit will be given for unsupported answers.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
If any question is not clear, ask for clariﬁcation. # Points Score 1 6 2 6 3 6 4 12 Σ 30 1. Newton’s law of cooling states that the temperature of an object changes at a rate
proportional to the diﬀerence between its temperature and the temperature of its
surroundings.
(a) (2 points) Let T = T (t) be the temperature of a cup of coﬀee sitting on a dining
room table as a function of time t in minutes. Let T0 be the temperature of the
dining room. Write a diﬀerential equation for T . (b) (4 points) Determine T = T (t) given that T (0) = 2T0 and T ′ (0) = −3. 2. (6 points) Solve the initial value problem
y′ + 2 y =
t
y(π) = 0 cos(t)
,
t2 t>0 (Hint: the diﬀerential equation is ﬁrstorder linear with integrating factor µ(t) = t2 .) 3. Consider the autonomous diﬀerential equation dy
= −y(2 − y)(5 − y).
dt (a) (2 points) Identify the equilibrium solutions. (b) (4 points) Use the plot of f (y) versus y below to determine which of the equilibrium solution(s) are asymptotically stable and which are asymptotically unstable.
Be sure to brieﬂy explain how you arrived at your answers.
f(y) 2 5 y 4. Find the general solution to each of the following diﬀerential equations.
(a) (4 points) y ′′ − 4 y ′ + 3 y = 0 (b) (4 points) y ′′ − 4 y ′ + 4 y = 0 (c) (4 points) y ′′ − 4 y ′ + 5 y = 0 ...
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 Spring '09
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 Math

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