Given the equation $y'+\alpha⋅y=\sin(\pi \cdot x)$ find all values of $\alpha$ for which there is no periodic solution.
What I have tried:
First I found $y$ and it is $$y = \frac{\alpha \cdot \sin(\pi \cdot x) - \pi \cdot \cos(\pi \cdot x ) }{\pi ^ 2 + \alpha^2 } + c \cdot e^{-αx} $$ where $c_1$ is constant. Now my idea is to find $y(x)-y(x+T) \neq 0 $ where T is the period. Is this the approach to take?