(a) Consider a linear system

\(u^\prime + c(t)u=0\),

where \(c\) is a periodic (complex-valued) function, that is, continuous on \(-\infty < t < \infty \). Show that if any nontrivial solution, then

\(u(t+\gamma ) = \exp \left [-\int_0^\gamma c(s)ds \right]u(t) ,\gamma >0 \),

for all \(t\), where \(\gamma\) is the period of \(c\).

(b) Under the above assumptions, show that any nontrivial solution \(u\) of \(u^\prime + c(t)u=0\), is periodic with period \(\gamma\) if \(\int_0^\gamma c(s)ds=0\).

(c) Assuming that \(c(t)\) is a constant \(c\), what conditions does \(c\) have to satisfy in order that any nontrivial solution of \(u^\prime +cu =0\) is periodic,

(i) with period \(\gamma\);

(ii) with period \(n\gamma\), where \(n\) is a positive integer.

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