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The present paper is devoted to studying the Hill-type formula and Krein-typetrace formula for ODE, which is a continuous work of our previous work forHamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formulaare given by Hill at 1877 and Krein in 1950's separately. Recently, we findthat there is a closed relationship between them \cite{HOW}. In this paper, wewill obtain the Hill-type formula for the $S$-periodic orbits of the firstorder ODEs. Such a kind of orbits is considered naturally to study thesymmetric periodic and quasi-periodic solutions. By some similar idea in\cite{HOW}, based on the Hill-type formula, we will build up the Krein-typetrace formula for the first order ODEs, which can be seen as a non-self-adjointversion of the case of Hamiltonian system.

Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs