NAME_____________________________________________________________________________________________________
MAYNARD
MATH 279
TEST 2
CHAPTER 4, 5
SAMPLE
1)
Given that y = c1e4x  c2ex is a two-parameter
family of solutions of y"  3y'  4y = 0 on the
interval (,), find a member of the family
satisfying the initial conditions
y(0) = 1, y'(0) = 2.
2)
Show by computing the Wronskian that the
given functions are linearly independent on
the indicated interval.
x1/2, x2 ; (0, )
3)
Verify that the given functions form a
fundamental set of solutions of the differential
equation on the indicated interval.
Form the general solution.
y"  2y'  5y = 0; ex cos 2x, ex sin 2x, (,)
4)
Find a second solution of the differential equation
using reduction of order. Assume an appropriate
interval of validity.
x2y"  7xy'  16y = 0;
y1 = x4
5)
Find a second solution of the differential equation
using reduction of order. Assume an appropriate
interval of validity.
(1  2x  x2)y"  2(1  x)y'  2y = 0;
6)
y1 = x  1
Find the general solution of the differential equation.
12y"  5y'  2y = 0
7)
Solve the differential equation subject to the initial
conditions.
2y"  2y'  y = 0, y(0) = 1, y'(0) = 0
8)
Find a differential operator that annihilates the function.
13x  9x2  sin 4x
9)
Solve the differential equation by the method
of undetermined coefficients using an annihilator.
y"  y'  12y = e4x
10)
Solve the differential equation by the method
of undetermined coefficients using an annihilator.
y"  2y'  5y = e x sin x
11)
Solve the differential equation by the method
of variation of parameters. State an interval on
which the general solution is defined.
y"  3y'  2y = 1/(1  e x )
12)
Solve the differential equation by the method
of variation of parameters. State an interval on
which the general solution is defined.
y"  2y'  y = e x ln x
13)
The period of free undamped oscillations of
a mass on a spring is /4 second. If the spring
constant is 16 lb/ft, what is the numerical
value of the weight?
14)
A 64-lb weight attached to the end of a spring
stretches it 0.32 ft. From a position 8 in. above
the equilibrium position the weight is given a
downward velocity of 5 ft/s.
a) Find the equation of motion.
b) What are the amplitude and period of motion?
c) How many complete vibrations will the weight
have completed at the end of 3 seconds?
d) At what time does the weight pass through the
equilibrium position heading downward for the
second time?
e) At what times does the weight attain its extreme
displacement on either side of the equilibrium
position?
f) What are the position, instantaneous velocity and
acceleration of the weight at t = 3 seconds?
A force of 2 lb stretches a spring 1 ft. A 3.2-lb wight
is attached to the spring and the system is then immersed
in a medium that imparts a damping force numerically
equal to 0.4 times the instantaneous velocity.
a) Find the equation of motion if the weight is released
from rest 1 ft above the equilibrium position.
b) Express the equation of motion in the form
15)

x(t)  Ae λt sin ω 2  λ 2 t  φ

c) Find the first time at which the weight passes through
the equilibrium position heading upward.
16)
A 10-lb weight attached to a spring stretches it 2 ft.
The weight is attached to a dashpot damping device
that offers a resistance numerically equal to  ( > 0)
times the instantaneous velocity. Determine the values
of the damping constant  so that the subsequent
motion is
a) overdamped
b) critically damped
c) underdamped.
17)
A mass m is attached to the end of a spring whose
constant is k. After the mass reaches equilibrium,
its support begins to oscillate vertically about a
horizontal line L according to a formula h(t).
The value of h represents the distance in feet
measured from L. Determine the differential
equation of motion if the entire system moves
through a medium offering a damping force
numerically equal to (dx/dt)
L
} h(t)
m
18)
A mass of 100 g is attached to a spring whose
constant is 1600 dynes/cm. After the mass reaches
equilibrium, its support oscillates according to the formula
h(t) = sin 8t, where h represents displacement from its
original position, in cm.
a) In the abscence of damping, determine the equation
of motion if the mass starts from rest from the
equilibrium position.
b) At what times does the mass pass through the
equilibrium position?
c) At what times does the mass attain its extreme
displacements?
d) What are the maximum and minimum displacements?
19)
Find the steady-state charge and the steady-state
current in an L-R-C series circuit when L = 1 henry,
R = 2 ohms, C = 0.25 farad, and E(t) = 50 cos t volts.
20)
Find the equation of motion describing the small
displacements (t) of a simple pendulum of
length 2 ft released at t = 0 with a displacement
of 1/2 radian to the right of the vertical and angular
velocity of 23 radians /s to the right. What are the
amplitude, period, and frequency of motion?
ANSWERS
1)
y = 3/5 e4x  2/5ex
1/2
2
3
3/2
a)
b)
c)
d)
19)
qp = 100/13 sin t  150/13 cos t;
ip = 100/13 cos t  150/13 sint
20)
(t) = 1/2 cos 4t  1/2 3 sin 4t;
amplitude = 1
period = /2
frequency = 2/
 0 on (0,  )
2)
W(x , x ) = /2 x
3)
The functions satisfy the differential equation
and are linearly independent on the interval, since the
Wronskian is never zero on the interval.
y = c1ex cos 2x  c2ex sin 2x
4)
y2 = x4 ln |x|
5)
y2 = x2  x  2
6)
y = c1e2x/3  c2ex/4
7)
y = ex/2 cos ( x /2 )  ex/2 sin ( x /2 )
8)
D3 (D2  16)
9)
y = c1e3x  c2e4x  1/7 xe4x
10)
y = ex (c1 cos 2x  c2 sin 2x)  1/3 ex sin x
11)
y = c1ex  c2e2x  (ex  e2x ) ln (1  ex)
( , )
12)
y = c1ex  c2xex  1/2 x 2 ex ln x  3/4 x 2 ex
(0 , )
13)
8 lb
14)
a) x(t) = 2/3 cos 10t  1/2 sin 10t
= 5/6 sin (10t  0.927)
b) 5/6 ft; /5
c) 15 cycles
d) 0.721 s
e) (2n  1)/20  0.0927, n = 0, 1, 2, ...
f) 0.597 ft; 5.814 ft/s; 59.702 ft/s2
15)
a) x(t) = e2t [ cos 4t  1/2 sin 4t]
b) x(t) = 1/25 e2t sin (4t  4.249)
c) 1.294 s
16)
a)  > 5/2
b)  = 5/2
c) 0 <  < 5/2
17)
mx" = k(x  h)  x' or
x"  (/m) x'  (k/m)x = (k/m)h(t) or
x"  2x'  2x = 2h(t)
x(t) = 2/3 sin 4t  1/3 sin 8t
t = n/4, n = 0, 1, 2, ...
t = /6  n/2, n = 0, 1, 2, ...
1
/23 cm, 1/23 cm
18)