On the set of positive real numbers, multiplication, represented by ⊕, is considered as an operation associated with the notion of sum, and the operation a ⨀ b = aln(b) represents the meaning of the traditional multiplication. The triple (R+, ⊕,⨀) forms an ordered and complete field in which derivative and integration operators are defined analogously to the Differential and Integral Calculus. In this article, we present the proportional arithmetic and we construct the theory of ordinary proportional differential equations. A proportional version of Gronwall inequality, Gompertz’s function, the q-Periodic functions, proportional heat, and wave equations as well as a proportional version of Fourier’s series are presented. Furthermore, a non-Newtonian logistic growth model is proposed.