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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?
69. 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
.
n4
n=1
7) (5 points)
∞ 1
√
.
n=2 n ln n
1 8) (5 points)
∞ sin3 4n
.
n4
n=1
9) (5 points)
∞ ln n
.
2
n=1 n
10. Consider the power series
∞ (−1)n−1
n=1 (3x)n
.
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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 Fall '07
 GuanYuShi
 Calculus, Equations

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