Theoretical question on cyclic function

[nhrock3]

a(x) is continues on R with cycle T ,a(x+T)=a(x)
u(x) is non trivial soluion of y'=a(x)y
$$\lambda=\int_{0}^{T}a(x)dx$$

which of the following claims is correct:

A. if $$\lambda>0$$ then $$\lim_{x\rightarrow\infty}u(x)=\infty$$
B. if $$\lambda=0$$ then u(x) is a cyclic function

i don't have the theorectical basis to solve it

 

[fzero]

The equation is separable. Write it as

$$\frac{y'}{y} = a(x)$$

and integrate both sides over one period.

 

[nhrock3]

how to integrate over a cycle
?
what to do after i integrate over a cycle
$$\int\frac{dy}{y}=\int a(x)dx$$

$$\ln y=\inta(x)dx$$

 

[fzero]

how to integrate over a cycle
?
what to do after i integrate over a cycle
$$\int\frac{dy}{y}=\int a(x)dx$$

$$\ln y=\inta(x)dx$$

Try

$$\int_{u(0)}^{u(T)}\frac{dy}{y}=\int_0^T a(x)dx$$