Contour integrals associated differential equations

For any n contour integrals y = cos n + 1 x ∮ C cosh ( z s ) ( sin z − sin x ) n + 1 d z , y = sin n + 1 x ∮ C cosh ( z s ) ( cos z − cos x ) n + 1 d z , s 2 = − λ , and their associated second-order ordinary differential equations which are not studied are given. Complete solution is given by calculating residues for n = 1 . The eigenvalues and eigenfunctions along with certain boundary conditions are explored for the first contour integral. The contour integral y ( x ) = cosh 3 x ∮ C cosh ( z s ) ( sinh z − sinh x ) 3 d z , s 2 = − λ , and its associated second-order ordinary differential equation which is also not studied are given. Explicit solution, eigenvalues, spectral function and eigenfunctions expansions are explored. These differential equations which do have solution in terms of the trigonometric functions which does not seem to have been explored and it is also one of the purposes of this paper to put them on record.