Homework Assignment 14 in MATH 308-SPRING 2015
due November 23, 2015
Topics covered : nonhomogeneous equations and systems: method of reduction of order; method of undetermined
coefficients (section 3.5 and 7.9) and its application to forced vibrations (section 3.8); use that the gravitational
acceleration g = 32 sf2t .
1. For each of the following equations write down the form in which a particular solution
should be found according to the method of undetermined coefficients (you do not need
to find the value of the undetermined coefficient/coefficients here):
(a) y 00 − 5y 0 − 15y = t3 − 4t2 + 5t − 7 + e5t (t2 − 1) + e3t t6 + 2t2 e−3t sin 5t;
(b) y 00 − 8y 0 + 16y = t4 e−4t + (3t4 + 7t)e4t cos 10t − (t2 + 5)e4t
3
3
(c) 4y 00 + 12y 0 + 25y = e− 2 t + 3e2t t4 sin 23 t − 5t4 e− 2 t cos 2t;
2. Find the general solution of the given equation and the system using the method of undetermined coefficients:
(a) y 00 + 6y 0 + 8y = 3e−2t + 2tet ;
(b) (bonus 30 points)
 0
 x1 =

4x1 + 31 x2 − 3e3t
x02 = 9x1 + 6x2 + 10e3t
3. A spring is stretched 0.5 in by a mass that weighs 8 lb. The mass is attached to a dashpot
mechanism that has a damping constant of 3 lb·s
f t and is acted on by an external force of
3 cos 5t − 2 sin 5t lb. Determine the steady state solution of this system, using the method
of undetermined coefficients.
1