Consider the ODE X'''' = λ X with boundary conditions X(0)=X'(0)=X(1)=X'(1)=0. Find the eigenvalues and eigenfunctions.
General solution of the ODE is:
X = Aexp(kx) + Bexp(-kx) + C sin(kx) + D cos(kx)
Case λ>0:
Let λ=k^4, with k>0
where k is solution cos(k) cosh(k) = 1. This gives all the positive eigenvalues.
To find the eigenfunctions, I got the following system of 4 equations in 4 unknowns by using the boundary condition:
0 = A+B+D
0 = A-B+C
0 = A exp(k) + B exp(-k) + C sin(k) + D cos(k)
0 = A exp(k) - B exp(-k) + C cos(k) - D sin(k)
Now how can I solve for A, B, C, D in the above system and find the eigenfunctions?
Attempt:
1st equation=> D = -A-B
2nd equation=> C = B-A
Put these into the 3rd and 4th equation, we get:
0 = A exp(k) + B exp(-k) + (B-A) sin(k) + (-A-B) cos(k)
0 = A exp(k) - B exp(-k) + (B-A) cos(k) - (-A-B) sin(k)
How should I continue??
Just wondering: In a system of 4 equations in 4 unknowns, is it POSSIBLE to have infinitely many solutions? or MUST the solution be unique?
(Since any nonzero multiple of an eigenfunction is again an eigenfunciton, I am expecting the solution of the system to have one arbitrary constant, i.e. infinitely many solutions, but is this possible in the above system?)
Any help is greatly appreciated!
[note: also under discussion in sos math cyberboard]