Solving Linear Differential System: Eigenvalues & Exact Solutions

[aaaa202]

I am studying a system described by a set of first order linear differential equations as can be seen on the attached picture. Now I know that to solve this analytically for a given N, N denoting the matrix size, one has to find the eigenvalues of the given matrix, which translates into finding the roots of an nth order polynomial, which is in general not possible.
But if you look at the matrix on the picture, which has a special structure, would it then be possible to find the exact eigenvalues of it and then find the exact solution of the system.
I should mention that the boundary conditions are simply:
x1(0) = 1, xn(0)=0, N≥n>1
If not by an eigenvalue method are there any other ways to find an exact solution?

 

[haruspex]

http://en.wikipedia.org/wiki/Tridiagonal_matrix might be of some help.
In your figure, the bottom right subscripts don't conform to the pattern of those at the top left.
Did you mean
n-1,n-2 n-1,n-1, n-1,n
n,n-2 0 0
?
Or did you intend the value in the last line to be the same as the one above it:
n-1,n-2 n-1,n-1, n-1,n
n-1,n-2 0 0
?

 

[aaaa202]

No the matrix differs does not follow the pattern in the first and last row.

 

[aaaa202]

But its still tridiagonal

 

[Ray Vickson]

No the matrix differs does not follow the pattern in the first and last row.

Your matrix is not square: it has n rows and (n+1) columns. Therefore, your DE system does not make sense.

 

[aaaa202]

Oh sorry the drawing is wrong. So it is a tridiagonal matrix as described in the link above but with a_NN zero and a_12 zero. Is it possible to solve the eigenvalue equation for such a matrix exactly?