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This is a translation of Kamke's ''Uber die selbstadjungierten Eigenwertaufgaben bei gewohnlichen linearen Differentialgleichungen. I. '', the first of three papers published in Mathematische Zeitschrift during during 1939 and 1940 on the eigenvalue problem L(u) = lambda M(u), where L and M are self-adjoint linear differential operators. These papers employ classical methods to develop detailed theorems on the spectra (lambda/sub p/) and give the appropriate extensions of the Rayleigh quotient and Courant max-min theorems, for the case where L is positive definite but M need not be. The direct approach taken by Kamke, and the clarity and concreteness of the results given, should make this material accessible to a much larger group of applied mathematicians than is the corresponding material of the linear-operator-in-Hilbert-space literature. (ERA citation 03:030010)