I was trying to solve the first order differential equation by fourier series:
$\dfrac{dx}{dt}-\beta x=\dfrac{1}{2}$
$$x(t)=\sum_{n=-\infty}^\infty x_ne^{int}$$
and $$x'(t)=\sum_{n=-\infty}^\infty in x_ne^{int}$$
by substituting x(t) and x'(t) in above equation. but after that i was stuck.
Where $\beta$ is complex variable. Any suggestions would be greatly appreciated.