Consider the homogeneous equation $ y'+a(x) y=0$, a is with period $\xi$ and continuous on $-\infty<x<\infty$.
(a) Show that there exist a constant c such that $y(x+\xi)=cy(x)$ for all x and $$c=exp \left(- \int_0^\xi \ a(t) dt \right) $$
(b) What condition must 'a' satisfy in order that there a exist a non- trivial solution of period $\xi$, of period $2\xi$? If 'a' is real valued, what is the condition?
(c) If a is constant, what must this constant be in order that a non-trivial solution of period 2$\xi$ exists ?
I have made an attempt at (a) but its leading to redundancy i think its time to ask for some help. Please help me if you can Thank.
To do part a, use integrating factors: http://en.wikipedia.org/wiki/Integrating_factor