Consider a non-homogeneous linear differential equation
\(u^\prime +c(t)u=d(t)\),
where \(c\) and \(d\) are given continuous periodic functions on \(-\infty<t< \infty \) of period \(\gamma\) and \(d(t) \neq 0\).
(a) Show that a solution \(u\) is periodic of period \(\gamma\) if and only if \(u(0)=u(\gamma)\).
(b) If there is no nontrivial periodic solution of period \(\gamma\) of the corresponding homogeneous equation, then prove that a unique solution of period \(\gamma\) for
\(u^\prime +c(t)u=d(t)\)
exists. Construct an example to illustrate (ii).
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