MATH 267 Section J
Fall 2012
PRACTICE FINAL EXAM
1. (20 points) Solve the initial value problem:
(a) y 0 =
1
xy
y(1) = 3.
(b) y 00 − 8y 0 − 20y = 0
y(0) = 5 y 0 (0) = 14.
2. (20 points) Find the general solution:
(a) y 0 +
y
=
x+1
00
x2 .
(b) y 000 + 6y + 9y 0 = x − 3ex .
3. (10 points) Solve by the Laplace transform method
y 00 + 6y 0 − 7y = u4 (t)
y(0) = 1
y 0 (0) = 2
4. (10 points) FindPthe recurrence relation and the first 5 nonzero terms in the power series
n
solution y(x) = ∞
n=0 an x of
y 00 + y 0 − xy = 0
y(0) = 2 y 0 (0) = 1
5. (10 points) Census Bureau estimates of the Unites States population are 281.4 million
for April 1, 2000 and 308.7 million for April 1, 2010. Assuming that population grows
at a rate proportional to the population, estimate the United States population on April
1, 2020 to the nearest million.
6. (10 points) Find the general solution of the system
−3 2
0
x =
x
2 −3
and sketch the phase portrait.
7. (10 points) Find all equilibrium solutions of the system
x0 = xy
y 0 = −x + 2y + 4
and classify each by type and stability.
8. (10 points) Show that {1 + x, ex } is a fundamental set for
xy 00 − (x + 1)y 0 + y = 0
Then use the variation of parameters method to find a specific solution of
xy 00 − (x + 1)y 0 + y = x2 ex
(Remember to write the equation so that the coefficient of y 00 is 1.)