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Unformatted text preview: Problems from Final exam, Spring 2011.
1. Solve the diﬀerential equations:
a) (10 points) y = xy + y (Find the general solution)
b) (10 points) x2 y = y ln y − y (Find the general solution)
c) (10 points) y = 2 sin x
y with initial condition y (π/2) = −4. 2. Solve the diﬀerential equation (ﬁnd the general solution):
a) (5 points) y + y − 20y = 0.
b) (5 points) y − 8y + 12y = 0.
3. (10 points) Solve the following initial value problem:
y + 6y + 9y = 0; y (1) = 5, y (1) = −2. 4. (8 points) A tank holds 60 pounds of salt dissolved in 120 gallons of
water. At time t = 0 brine containing 3 pounds of salt per gallon begins to
ﬂow into the tank at the rate of 2 gallons per minute and continues to do so.
Beginning at the same time the perfectly mixed solution ﬂows out of a tap
at the bottom of the tank at the same rate.
How much salt is dissolved in the tank after 60 minutes?
5. (7 points) A cup of coﬀee has temperature 90 degrees Centigrade in a
room kept at 20 degrees Centigrade. After half an hour the coﬀee cools to
80 degrees Centigrade.
What is the temperature after another half hour?
6-9. Test for convergence (no credit for answer without a correct complete justiﬁcation). If you use a test, you must show how it works, i.e.
you can’t just say :”comparison test” or ”integral test”, you have to make
the comparison or compute the appropriate integral.
6) (5 points)
1 + sin4 4n
7) (5 points)
n=2 n ln n
1 8) (5 points)
∞ sin3 4n
9) (5 points)
∞ ln n
10. Consider the power series
n a) (4 points) Find the radius of convergence.
b) (4 points) Find the interval of convergence.
11. (7 points)
a) Find the Taylor series for
f (x) = 1
x2 at a = 1. b) What is the radius of convergence? 2 ...
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