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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?

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?