MATH 266 Section A1
Spring 2013
PRACTICE FINAL EXAM
10 points each
1. Find the general solution of y 0 = 2y 2 + 3.
2. Solve the initial value problem y 0 +
y
x+1
= x2
y(1) = 2.
3. Find the general solution of y 000 + 6y 00 + 9y 0 = t − 3et .
4. Solve the initial value problem y 00 − 8y 0 − 20y = 0
5. Find the general solution of y 0 =
y(0) = 5 y 0 (0) = 14.
y 2 −xy
x2
6. 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.
7. Find the solution of
2 2
x =
x
1 3
0
8. Let
0
x(0) =
2


1 −1 −1
0
A = 1 1
3 0
1
Find a fundamental set for x0 = Ax.
9. Find all equilibrium solutions of the system
x0 = xy
y 0 = −x + 2y + 4
and classify each by type and stability.
10. Show that {1 + t, et } is a fundamental set for
ty 00 − (t + 1)y 0 + y = 0
Then use the variation of parameters method to find a particular solution of
ty 00 − (t + 1)y 0 + y = t2 et