Before starting the process, you should have some idea of the form of the solution you are looking for. Some experience helps here.
For example, for unsteady heat conduction in a bar of length ,the temperature u would be written
For a uniform bar, you will find sines and/or cosines for the functions fn. In that case the above expansion for u is called a Fourier series. In general it is called a generalized Fourier series.
After the functions fn have been found, the ``Fourier coefficients'' Cn can simply be found from substituting the expression for u in the given P.D.E. and initial conditions. (The boundary conditions are satisfied when you choose the eigenfunctions fn.)
If the problem was axially symmetric heat conduction through the wall of a pipe, the temperature would still be written
For heat conduction through a pipe wall without axial symmetry, the temperature would be written
(For steady heat conduction, the coordinate ``t'' might actually be a second spatial coordinate. For convenience, we will refer to conditions at given values of t as ``initial conditions'', even though they might physically really be boundary conditions.)