MATH 308, Spring 2016
Take Home QUIZ # 7
Due: Thursday 03 / 24 / 2016
Print name (LAST, First):
SECTION #:
INSTRUCTOR: Dr. Marco A. Roque Sol
THE AGGIE CODE OF HONOR
"An Aggie does not lie, cheat, or steal, or tolerate those who do." By signing below, you indicate that all
work is your own and that you have neither given nor received help from any external sources.
SIGNATURE:
1. (10 pts.) Find the solution to the given initial problem
y 00 + 2y 0 + 5y = 4e−t cos(2t);
y(0) = 1, y 0 (0) = 0
.
2. (10 pts.) Find the general solution to
00
N
X
2
y +λ y =
.
where λ > 0 and λ 6= mπ f or m = 1, 2, ..., N .
am sin(mπt)
m=1
3. (10 pts.) Verify that the given functions y1 and y2 satisfy the corresponding homogeneous equation; then nd a
particular solution of the given nonhomogeneous equation using the method of undetermined coecients.
ty 00 − (1 + t)y 0 + y = t2 e2t ;
t > 0;
y1 = 1 + t.
y2 = et
4. (10 pts.) Use the method of variation of parameters to nd a particular solution of the given dierential equation
and then nd the general solution of the ODE.
a) y 00 + y = tan(t)
.
b) y 00 − 2y 0 + y =
.
et
(1 + t2 )
5. ( 10 pts. ) A mass-spring system is modeled by the initial value problem
2u00 + 2u = cos(0.8t);
u(0) = 0, u0 (0) = 0
.
Describe physically the problem and nd the explicit solution. Using MATLAB to graph the solution of this
problem.
6. ( 20 pts. ) Determine ω0 , R, and δ so as to write the given expression in the form u = Rcos(ω0 t − δ) .
a) u = 4cos(3t) − 2sin(3t)
.
b) u = −2cos(πt) − 3sin(πt)
.
Using MATLAB to graph the solutions of these problems.
7. ( 10 pts. ) A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring
a distance of 1 in, and then set in motion with a downward velocity of 2 ft/s, and if there is no damping, nd
the position u of the mass at any time t. Determine the frequency, period, amplitude, and phase of the motion.
Using MATLAB to graph the solution of this problem u = u(t).
8. ( 10 pts. )A spring is stretched 10 cm by a force of 3 N. A mass of 2 kg is hung from the spring and is also
attached to a viscous damper that exerts a force of 3 N when the velocity of the mass is 5 m/s. If the mass is
pulled down 5 cm below its equilibrium position and given √
an initial downward velocity of 10 cm/s, determine
2
and the ratio of µ to the natural frequency
its position u at any time t. Find the quasi frequency µ = 4m−γ
2m
of the corresponding undamped motion. Using MATLAB to graph the solution of this problem u = u(t).