Name________________________
Student I.D.___________________
Math 2250-1
Quiz 9 Solutions
November 2, 2012
1) Consider the following mechanical oscillation differential equations. In each case answer the following
questions:
(i) What is the undetermined coefficients "guess" for the particular solution xP t ? (Do NOT try to find
the precise particular solution, just its form.)
(ii) What physical phenomenon will be exhibited by the general solutions to this differential equation?
Each problem is worth three points, with one free point.
1a) x## t C 4 x t = cos 2 t
(i) For the homogeneous DE, p r = r2 C 4 = r C 2 i 1 r K 2 i 1 . Since the roots r =G 2 i are
associated with the real solutions cos 2 t , sin 2 t . Thus the undetermined coefficients guess is
xP t = t Acos 2 t C Bsin 2 t .
(ii) Resonance.
1b)
x## t C 5 x t = cos 2 t .
(i) Since the natural angular frequency is w0 = 5 s 2 and since only even derivatives occur on the leftside, there will be an undetermined coefficients solution of the form
xP t = A cos 2 t .
You could also guess a particular solution of the form
xP t = A cos 2 t C B sin 2 t
(it would turn out that B = 0).
(ii) Since the natural angular frequency w0 =
solutions will exhibit beating.
5 z 2.4 is relatively close to 2 (in ratio), the general
1c) x## t C 0.2 x# t C 4 x t = cos 2 t .
(i) Since the characteristic polynomial has complex roots which are NOT G 2 i , we guess
xP t = A cos 2 t C B sin 2 t .
(ii) Because the damping is small (c = 0.2 and the angular frequency of the forcing function (w = 2 is
close to, well actually equals, the natural frequency for the undamped problem, the solution will exhibit
practical resonance.