In the spring-mass system of Problem \(31,\) suppose that the spring force is
not given by Hooke's law but instead satisfies the relation
$$
F_{s}=-\left(k u+\epsilon u^{3}\right)
$$
where \(k>0\) and \(\epsilon\) is small but may be of either sign. The spring is
called a hardening
spring if \(\epsilon>0\) and a softening spring if \(\epsilon<0 .\) Why are these
terms appropriate?
(a) Show that the displacement \(u(t)\) of the mass from its equilibrium
position satisfies the
differential equation
$$
m u^{\prime \prime}+\gamma u^{\prime}+k u+\epsilon u^{3}=0
$$
Suppose that the initial conditions are
$$
u(0)=0, \quad u^{\prime}(0)=1
$$
In the remainder of this problem assume that \(m=1, k=1,\) and \(\gamma=0\).
(b) Find \(u(t)\) when \(\epsilon=0\) and also determine the amplitude and period
of the motion.
(c) Let \(\epsilon=0.1 .\) Plot (a numerical approximation to) the solution.
Does the motion appear to be periodic? Estimate the amplitude and period.
(d) Repeat part (c) for \(\epsilon=0.2\) and \(\epsilon=0.3\)
(e) Plot your estimated values of the amplitude \(A\) and the period \(T\) versus
\(\epsilon\). Describe the way in which \(A\) and \(T\), respectively, depend on
\(\epsilon\).
(f) Repeat parts (c), (d), and (e) for negative values of \(\epsilon .\)
