8 1. Differential Equations and Their Solutions can we say about the existence and uniqueness of solutions to such initial value problems?” We will discuss this central question in detail below, along with important related questions such as how solutions of an IVP change as we vary the initial condition and the vector field. In order not to over-burden the exposition, we will leave many details of proofs to be worked out by the reader in exercises (with liberal hints). Fully detailed proofs can be found in the appendices and various references. First let us make precise the definition of a solution of the above initial value problem: it is a differentiable map x of some open interval I containing t0 into Rn such that x(t0) = x0 and x (t) = V (x(t),t) for all t in I. We first consider uniqueness. The vector field V : Rn × R Rn is called continuously differentiable (or C1) if all of its components Vi(x1,... , xn,t) have continuous first partial derivatives with respect to x1,... , xn,t, and more generally V is called Ck if all partial deriva- tives of order k or less of its components exist and are continuous. 1.1.1. Uniqueness Theorem. Let V : Rn × R Rn be a C1 time-dependent vector field on Rn and let x1(t) and x2(t) be two solutions of the differential equation dx dt = V (x, t) defined on the same interval I = (a, b) and satisfying the same initial condition, i.e., x1(t0) = x2(t0) for some t0 I. Then in fact x1(t) = x2(t) for all t I. Exercise 1–1. Show that continuity of V is not sufficient to guar- antee uniqueness for an IVP. Hint: The classic example (with n = 1) is the initial value problem dx dt = x and x(0) = 0. Show that for each T 0, we get a distinct solution x T (t) of this IVP by defining x T (t) = 0 for t T and x T (t) = 1 4 (t T )2 for t T . But what about existence? 1.1.2. Local Existence Theorem. Let V : Rn × R Rn be a C1 time-dependent vector field on Rn. Given p0 Rn and t0 R, there is a neighborhood O of p0 and an 0 such that for every p
Previous Page Next Page