MATH 215/255
Fall 2014
Assignment 5
§2.5, §2.6
Solutions to selected exercises can be found in [Lebl], starting from page 303.
• 2.5.7: a) Find a particular solution of y 00 − 2y 0 + y = ex using the method of variation
of parameters.
b) Find a particular solution using the method of undetermined coefficients.
• 2.5.9: For an arbitrary constant c find a particular solution to y 00 − y = ecx . Make
sure to handle every possible real c.
• 2.5.101: Find a particular solution to y 00 − y 0 + y = 2 sin(3x).
• 2.5.103: Solve y 00 + 2y 0 + y = x2 , y(0) = 1, y 0 (0) = 2.
• 2.6.1: Derive a formula for xsp when the equation is mx00 + cx0 + kx = F0 sin(ωt).
• 2.6.2: Derive a formula for xsp when the equation is mx00 + cx0 + kx = F0 cos(ωt) +
F1 cos(3ωt).
• 2.6.3: Fix parameters F0 , k, m > 0. Consider the equation mx00 + cx0 + kx =
F0 cos(ωt). For what values of c (in terms of F0 , k, m) will there be no practical
resonance? In other words, for what values of c is there no maximum of C(ω) for
ω > 0?
• 2.6.4: Fix parameters F0 , c, k > 0. Consider the equation mx00 +cx0 +kx = F0 cos(ωt).
For what values of m (in terms of F0 , c, k) will there be no practical resonance?
• 2.6.101: A mass m = 4 is attached to a spring with k = 4 and a damping constant
c = 1. Suppose that F0 = 2. Using the forcing function F0 cos(ωt), find the ω that
causes practical resonance and find the amplitude.
• 2.6.102: Derive a formula for xsp when the equation is mx00 +cx0 +kx = F0 cos(ωt)+K,
where K is some constant.