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
[TEX]\lambda=\int_{0}^{T}a(x)dx[/TEX]

which of the following claims is correct:

A. if [TEX]\lambda>0 [/TEX] then [TEX] \lim_{x\rightarrow\infty}u(x)=\infty [/TEX]
B. if [TEX]\lambda=0 [/TEX] 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

[tex]\frac{y'}{y} = a(x)[/tex]

and integrate both sides over one period.

 

[nhrock3]

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

[tex]\ln y=\inta(x)dx[/tex]

 

[fzero]

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

[tex]\ln y=\inta(x)dx[/tex]

Try

[tex]\int_{u(0)}^{u(T)}\frac{dy}{y}=\int_0^T a(x)dx[/tex]