Question

Show by verifying the hypothesis of the Existence and Uniqueness Theorem that the initial value problem x dot = 1+x^2, x(0) = 0 has a unique solution. Find the solution. what is the maximal interval of definition of the solution? I'll put what I think is the answer as a response. Could you tell me if I took the right approach?

Answer 1

Because f: R-> R is continuous, then for any x knot which is an element of R, there is an interval (alpha,beta) containing 0 and there is a solution x(t) of x dot = f(x). The limit exist, therefore a solution exists. Because f is differentiable and f' is continuous, then x(t) is unique.

Answer 2

The solution is 0 and the maximal interval of definition is (-infinity,infinity)