Summary: Ordinary differential equations, initial conditions and integral equations.
We fix a positive integer n. The first time you read this I suggest you take n = 1.
Suppose is a subset of R × Rn
,
F : Rn
and F is continuous. We say x is a solution of the ordinary differential equa-
tion
(ODE) x (t) = F(t, x(t))
on I if
(i) I is an open interval in R and x : I Rn
;
(ii) x is differentiable at each point of I;
(iii) {(t, x(t)) : t I} and
(iv) (ODE) holds for each t I.
Let us make the following very useful observation. Suppose
(v) F is continuous;
(vi) (i) and (iii) hold and x is continuous;
(vii) t0 I, (t0, x0) and x satisfies the initial condition
(IC) x(t0) = x0.
Then x is a solution of (ODE) on I satisfying (IC) if and only if x satisfies the

Source: Allard, William K. - Department of Mathematics, Duke University

Collections: Mathematics