Math 340
Final
Professor Carlson
SHOW ALL YOUR WORK
Problems are worth 10 points each.
1. a) Find the general solution of
x2 + 5
dy
=
.
dx
y+3
b) Find the solution satisfying y(1) = 2.
2. Find the general solution of
a) y ′ + t2 y = 0.
1
b) y ′ + y = t3 .
t
3. Consider the differential equation
dy
= f (y),
dt
f (y) = y(y − 1)(y − 2).
Sketch the graph of f (y) vs. y. Find the equilibrium solutions and determine whether the equilibrium solutions are unstable or asymptotically
stable. Sketch the solutions y(t) vs. t.
4. a) Find distinct solutions y1 = er1 t and y2 = er2 t of the equation
y ′′ − 5y ′ + 4y = 0.
b) Compute the Wronskian of the solutions y1 (t) and y2 (t). What can
you conclude about linear dependence or independence of these solutions?
c) Find the solution satisfying
y(0) = 2,
y ′ (0) = 5.
5. Find the general solution of the equation
y ′′ + 4y = sin(t).
6. The variation of parameters formula gives a particular solution of
y ′′ + p(t)y ′ + q(t)y = g(t)
of the form
yp (t) = −y1 (t)
Z
t
t0
y2 (s)g(s)
ds + y2 (t)
W (y1 , y2 )(s)
Z
t
t0
y1 (s)g(s)
ds.
W (y1 , y2 )(s)
Consider the equation
y ′′ − y = e−2t ,
(A).
a) Show that y1 (t) = et and y2 (t) = e−t are independent solutions of
the corresponding homogeneous equation.
b) Find a particular solution of (A) by using the method of variation
of parameters with t0 = 0.
c) Find the general solution of (A).
7. Use the power series method to find solutions
y=
∞
X
ak xk
k=0
of
y ′ − 2y = 0
by
a) finding the recursion formula for the coefficients ak , and
b) finding an explicit formula for the coefficients ak if y(0) = 1.
8. Consider the differential equation
x2 y ′′ + αxy ′ + βy = 0,
x > 0.
(B)
a) Show that y = xr will be a solution of the equation if
r(r − 1) + αr + β = 0.
b) Find the general solution of
x2 y ′′ + xy ′ − 4y = 0,
x > 0.
9. The general form of the equations for the displacement xj of two
masses mj in line with three springs is
d2 x1
m1 2 + (k1 + k2 )x1 − k2 x2 = 0.
dt
d2 x2
m2 2 − k2 x1 + (k2 + k3 )x2 = 0.
dt
Suppose the masses are equal, m = m1 = m2 and the spring constants
satisfy k2 = k and k1 = k3 = αk for some constant α > 0.
(a) By adding and subtracting the equations, find second order equations for y = x1 + x2 and z = x1 − x2 . Find the general solutions for y and
z.
(b) Describe the solutions x1 and x2 if the initial displacements and
velocities are equal, that is
x1 (0) = x2 (0),
x′1 (0) = x′2 (0).
10. Suppose (in the absence of birds) the population of mosquitoes increases at a rate proportional to the current population, and the population
doubles each week.
a) What differential equation describes the mosquito population?
b) If there are 200 mosquitoes initially, and birds eat 20 mosquitoes per
day, what differential equation describes the mosquito population? What
is the population of mosquitoes as a function of time?