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Q3.21 A strained-coordinate problem III. See Q3.20; follow this same procedure for

as Show that as where is the appro-
priate solution of the equation (which does possess one real
root).
Q3.22 A strained-coordinate problem IV. See Q3.19; follow this same procedure for the
problem

as (You may observe that this problem can be solved exactly.)
Q3.23 A strained coordinate problem V. See Q3.19; follow this same procedure for the
problem

as If the boundary condition had been with the same
domain, briefly investigate the nature of this new problem.
Q3.24Duffing’s equation. The equation for the motion of a simple pendulum, without
the approximation for small angles of swing, takes the form

If x is small, and we retain terms as far as we obtain an equation like

this is Duffing’s equation (Duffing, 1918) which was introduced to improve the
approximation for the simple pendulum (without the complications of working
with sin x). Seek a solution of this equation, for and

by using a strained-coordinate formulation:

where the are constants. Determine the solution as far as terms in
choosing each in order to ensure that the solution is periodic.