I am working through the solution to a problem which I believe involves Stokes' oscillating boundary, the problems is as follows: There is an incompressible fluid with a boundary at $y=o$. The boundary is oscillating in the x direction with velocity $U\cos{\omega t}$. There is no applied pressure gradient. Assuming a flow of $\textbf{u}=(u(y, t), 0, 0)$, first show that $\textbf{u}$ satisfies, $$\partial_tu = \nu\partial^2_yu$$ Then find a solution of the form $u = \Re\lbrace{f(y)e^{i\omega t}\rbrace}$.
I managed to do this by making $f(y)e^{i\omega t}$ satisfy the (heat) equation above and adding a condition where $f(y)$ must turn to zero as y tends to infinity (so there are no disturbances far from the oscillating boundary).
What I want to know is why this works? I just happened to try it to see if it would work out and it does. Is there some theorem that states for a complex valued function satisfying a PDE, its real and imaginary parts also satisfy the PDE?
Perhaps this is very cumbersome to explain, in which case I appologise profusely, but perhaps there is some book or webpage you could point me in the direction of that would explain this?
Thank you very much.