MATH 267 Section P
Fall 2015
EXAM 1
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Ten points each
1. Find the solution of the initial value problem
1
y 0 = y cos (3x)
2
π
y( ) = 2
3
Solution:
y(x) = 2e
sin 3x
6
2. Find the general solution of
y0 +
Solution:
1
y(x) =
(x − 1)2
2
y=x
x−1
x4 2x3 x2
−
+
+C
4
3
2
3. Find any values of m for which y(x) = xm is a solution of the differential
equation
12
y 00 − 2 y = 0
x
Solution: m = −3 or m = 4
4. Show that the following equation is exact, and solve the initial value
problem.
4x3 y 2 + (2x4 y + ey )y 0 = 0
y(0) = 2
MATH 267 Section P
Fall 2015
EXAM 1
Solution:
x4 y 2 + ey = e2
5. Find the general solution of
y 0 = (2x + y − 5)3 − 2
(Suggestion: make a substitution)
Solution:
y(x) = √
±1
− 2x + 5
C − 2x
6. A cup of coffee at a temperature of 190◦ F is placed in a room held at
constant temperature 75◦ F. If after 20 minutes the coffee is observed to
have a temperature of 160◦ F, how long more will it take to reach 120◦ F?
Solution: The cup reaches temperature 120◦ F at time
t = 20
ln 23/9
≈ 62.08
ln 23/17
minutes after the cup is first placed in the room. Thus it take approximately 42.08 more minutes after reaching 160◦ F.
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