# Existence and uniqueness of solutions for IVPs

What is the best statement that you can make about the existence and uniqueness of the solutions of the following initial value problems? (Please see attached).

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#### Solution Preview

Hello there,

Recall Picard's Existence Theorem which says that when y' = f (y,t) where f is continuous then if the Lipschitz condition Modulus[ f(y1,t) - f(y2,t)] less than or equal to K Modulus[y1 - y2] is satisfied where

{(y,t): Modulus[y - y0] < b, Modulus[t - t0] < a} then y has a unique solution in the locality

Modulus[t - t0] < d = Min [a, b/Sup[f]]

This just means that f is well-behaved in the neighbourhood of a reference point.

For equation 1, f = Sin[ty] + 1/t and y(1) = 2.

f and its partial derivative with respect to y ( t Cos [ty]) exist and are continuous in the region around t = 1, so at least in that locality there is a unique solution. It looks that the unique ...

#### Solution Summary

This shows how to determine the existence and uniqueness of solutions to initial value problems.