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# Existence and uniqueness of solutions for IVPs

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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).

https://brainmass.com/math/algebra/existence-uniqueness-solutions-ivps-44056

#### 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.

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