ABSTRACT
Definition 57. Let 〈t, y〉 7→ f(t, y) be a function of two variables, defined for all 〈t, y〉 ∈ I × J . A differential equation is an equation of the form
y′ = f(t, y).
The unknown y is a function t 7→ y(t); it is a solution of the differential equation above if y′(t) = f(t, y(t)) holds for all t ∈ I. We restrict our attention to the case when 〈t, y〉 7→ f(t, y) is continuous in each of its variables, that is, for each fixed t in I the function y 7→ f(t, y) is continuous on J and for each fixed y ∈ J the function t 7→ f(t, y) is continuous on I. We consider only solutions whose derivatives are continuous on I (smooth solutions).
