[SOLVED] Amath 568 advanced differential equations homework 4

1. Consider the weakly nonlinear oscillator:
d
2
y
dt2
+ y + ϵy5 = 0
with y(0) = 0 and y
′
(0) = A > 0 and with 0 < ϵ ≪ 1.
(a) Use a regular perturbation expansion and calculate the first two terms.(b) Determine at what time the approximation of part (a) fails to hold.(c) Use a Poincare-Lindstedt expansion and determine the first two terms and
frequency corrections.(d) For ϵ = 0.1, plot the numerical solution (from MATLAB), the regular expansion solution, and the Poincare-Lindstedt solution for 0 ≤ t ≤ 22. Consider Rayleigh’s equation:
d
2
y
dt2
+ y + ϵ
”
−
dy
dt +
1
3

dy
dt 3
#
= 0
which has only one periodic solution called a ”limit cycle” (0 < ϵ ≪ 1). Given
y(0) = 0
and
dy(0)
dt =
(a) Use a multiple scale expansion to calculate the leading order behavior(b) Use a Poincare-Lindsted expansion and an expansion of A = A0 + ϵA1 + . . .
to calculate the leading-order solution and the first non-trivial frequency shift
for the limit cycle.(c) For ϵ = 0.01, 0.1, 0.2 and 0.3, plot the numerical solution and the multiple
scale expansion for 0 ≤ t ≤ 40 and for various values of A for your multiple
scale solution. Also plot the limit cycle solution calculated from part (b).(d) Calculate the error
E(t) = |ynumerical(t) − yapproximation(t)|
as a function of time (0 ≤ t ≤ 40) using ϵ = 0.01, 0.1, 0.2 and 0.3