A model problem about singular limits

[glebovg]

Homework Statement

Consider the linear differential equation εx'' + x' + x = 0, subject to the initial conditions x(0) = 1, x'(0) = 0.

a) Solve the problem analytically for all ε > 0.
b) Now suppose ε≪1. Show that there are two widely separated time scales in the problem, and estimate them in terms of ε.
c) Graph the solution x(t) for ε≪1, and indicate the two time scales on the graph.
d) What do you conclude about the validity of replacing εx'' + x' + x = 0 with its singular limit x' + x =0?
e) Give two physical analogs of this problem, one involving a mechanical system, and another involving an electrical circuit. In each case, find the dimensionless combination of parameters corresponding to ε, and state the physical meaning of the limit ε≪1.

The attempt at a solution

The corresponding characteristic equation is ελ^2 + λ +1 = 0 and λ = (-1 ± √1 - 4ε)/2ε. Since ε > 0 we have 1 - 4ε < 0 and two complex roots.
So x = Ae^(r*t)*cos(ω*t) + Be^(r*t)*sin(ω*t), where r = -1/2ε and ω = (|√1-4ε|)/2ε.
Using x(0) = 1 and x'(0) we get A = 1 and B = - r/ω.
Thus x = e^(r*t)*cos(ω*t) - (r/ω)e^(r*t)*sin(ω*t).

 

[lippyka]

b) Since ε is small, we have r ≈ 0 and ω ≈ 1. Thus, two time scales are t ≈ 0 and t ≈ 1/ε.c) Graph of x(t) for ε≪1[graph]The two time scales can be seen as the points where the graph intersects the x-axis.d) The singular limit is not valid since x(t) does not remain constant in the limit ε≪1.e) Mechanical system: ε corresponds to the damping coefficient in the equation of motion, and ε ≪ 1 implies that the damping is negligible.Electrical circuit: ε corresponds to the ratio of the capacitance to the inductance, and ε ≪ 1 implies that the inductive reactance is much greater than the capacitive reactance.