Differential equations with contour integrals

For any n, the contour integral [image omitted] is associated with the differential equation [image omitted] Explicit solutions are calculated by residues for n = 0, 1, 2. Eigenvalues along with certain boundary conditions and the asymptotics of the solutions with Im () 0 are explored for n = 1, 2. There is, in particular, one differential equation that does have a 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 article to put it on record.