Thanks for the lecture, it's great. Can I check if the solution to the ODE is correct? Seems like it should be sin/cos of sqrt(lambda)* x (instead of lambda* x). If there is in fact a typo, then the eigenvalues should also be (n pi / l ) ^2, that's the only result that is affected I think.
Generally and imprecisely, it is possible to write lambda instead of sqrt(lambda), to have simple and pretty form as it can happen throughout all Math, but new lambda is not eigenvalue anymore. And the result should be then eigenvalue=lamda^2
Nice example of generalized Gramm-Schmidt orthogonalization in new transformed space expressed in natural basis. Then regarding to that basis we need only to add lamda.
Yeah, that's great, but in order to do all that, we need to have the eigenfunctions and eigenvalues first. For matrices, there's at least an algorithm for that. But how are we supposed to do that with differential operators? :q
You mentioned that using eigenvalues is an easier way of getting Ax=b solution and explained that we should build the sigma expression from right-hand-side to calculate u(x); however, when it comes to the example, you just say the equation looks familiar and the answer is sin(x)+cos(x). If we want to guess the solution at the end, what is the point of using eigen values/functions?
He used the eigenvalue problem Ax=cx to find the eigenfunctions. The solution to this problem is well known in this case. Then he uses the normalized eigenfunctions to find the solution to Ax=b by expanding x in terms of the normalized eigenfunctions. In general, this technique is based on the idea of “using a known result “ (i.e. the solution to the eigenvalue problem) or “solving a simpler problem “ (i.e. the eigenvalue problem). See Larson or Poyas on mathematical problem solving. It will change your life.
Yeah, decomposition with inner products is the obvious part. Finding the eigenfunctions and eigenvalues of a particular differential operator (especially when the coefficients are not constant) is the hard part, and no one seems to be explaining that. Without it, all the rest is pretty much useless.
Kinda but not really, Fourier uses complex exponential basis. However the analog of those elements here are the eigenfunctions of the operator itself, solution is written as a sum of the eigenfunctions it is therefore operator dependent what your expansion is.
