MATH 267
FALL 2004
REVIEW PROBLEMS
1. a) Find the general solution of y 0 = y(y − 2).
b) Find all equilibrium solutions and determine whether they are
stable or unstable.
2. Solve the initial value problem
y 00 − 2y 0 − 15y = 0
y(0) = 2 y 0 (0) = 9
3. Show that the equation
(2xey − e−x ) + (x2 ey − 2y)y 0 = 0
is exact. Find the solution satisfying y(1) = 0.
4. Find the general solution of
y 00 + 4y = 8x2 + 2e−3x
5. Find a fundamental set for the Euler equation x2 y 00 −2xy 0 +2y = 0.
Use this fundamental set and the variation of parameters method to
find a particular solution of x2 y 00 − 2xy 0 + 2y = x3 ex . (Remember to
rewrite the equation so that the coefficient of y 00 is 1.)
6. Find the general solution of the Bernoulli equation
y0 + y =
(Hint: substitute v = y 2 .)
ex
y
7. Find the general solution of y 0 +
2
y = x.
x−1
8. A spherical raindrop evaporates at a rate proportional to its surface
area. Write a differential equation for the volume of the raindrop as a
function of time, and find the general solution of this equation. (Hint:
you will need to express the volume as a function of the surface area.)
9. Let f (t) =
2−t
0
0<t<2
t>2
a) Find the Laplace transform F (s) = L{f (t)}.
b) Solve for x(t) by the Laplace transform method if
x00 + 9x = f (t) x(0) = 1 x0 (0) = 6
10. Census Bureau estimates of the United States population are
248.7 million for April 1, 1990 and 281.4 million for April 1, 2000.
Assuming that the rate of population growth is proportional to population present, estimate the United States population on April 1, 2005
to the nearest million.