It is not always possible on separation of variables to separate initial or boundary conditions in a condition on one of the two functions. We can either map the problem into simpler ones by using superposition of boundary conditions, a way discussed below, or we can carry around additional integration constants.
Let me give an example of these procedures. Consider a vibrating string attached to two air bearings, gliding along rods 4m apart. You are asked to find the displacement for all times, if the initial displacement, i.e. at s is one meter and the initial velocity is .
The differential equation and its boundary conditions are easily written down,
Question: What happens if I add two solutions and of the differential equation that satisfy the same BC’s as above but different IC’s,
Answer: = , we can add the BC’s.
If we separate variables, , we find that we obtain easy boundary conditions for ,
(5.42) |
but we have no such luck for . As before we solve the eigenvalue equation for , and find solutions for , , and . Since we have no boundary conditions for , we have to take the full solution,
and thus
(5.44) |
Now impose the initial conditions
(5.45) |
which implies
,
.
(5.46) |
This is the Fourier sine-series of , which we have encountered before, and leads to the coefficients and if is odd and zero otherwise.
So finally
(5.47) |