Nov 27, 2022 · Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems. In this section we state such a condition and illustrate it with examples.

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or …

Proof of the uniqueness part of the theorem. Here we show that the problem (3.1) (and thus (1,1)) has at most one solution (we have not yet proved that it has a solution at all).

The theorem below shows that one can, under the right conditions, assert that a DE has a unique solution, even if the solution can’t be written down in closed form.

The existence and uniqueness theorem for initial value problems of ordinary differential equations implies the condition for the existence of a solution of linear or non-linear initial value problems and ensures the uniqueness of the obtained solution.

One of the most important theorems in Ordinary Di↵erential Equations is Picard’s Existence and Uniqueness Theorem for first-order ordinary di↵erential equations. Why is Picard’s Theorem so important? One reason is it can be generalized to establish existence

Hence, the theorems guarantee existence of unique solution in jxj 1=2, which is much smaller than the original interval jxj 100. Since, the above equation is separable, we can solve it exactly and nd y(x) = tan(x).

The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.

We will now see that rather mild conditions on the right hand side of an ordinary di erential equation give us local existence and uniqueness of solutions. De nition 4.1. Let f : D ! Rneb a oninuousc function de ned in the open set D R Rn. We say that f is locally Lipschitz in the Rn.

We introduce a version of Existence and Uniqueness theorem for rst order ODEs, which gives su cient conditions for a solution of an initial value problem to exist locally (i.e., only nearby the initial point).