## Fourier Series and Linear Differential Equations

Hello!

I have this easy LDE:

x'(t) = cos(t) x(t)
x(0) = 1

The solution of this equation is:

x = exp(sin(t))

This is a periodic function, which can be written as a Fourier Series.
Let's imagine that we do not know the solution but we know that it is periodic and that it can be written as a Fourier Series. Can we solve the equation using the same idea when using Power Series but with Fourier Series? How would it work?

Thank you!