In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article . As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article  and the article . And applying the theorems of Riemann integral introduced in the article , we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in .