where

Why doesn't this equation satisfy the hypothesis of the uniqueness theorem?

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- Mar 26th 2009, 06:04 PMLenDifferential Equations
where

Why doesn't this equation satisfy the hypothesis of the uniqueness theorem? - Mar 26th 2009, 11:55 PMchisigma
For semplicity let's suppose that , so that the equation is...

(1)

Writing a generic first order equation in the form...

(2)

... with 'initial condition' , it admits one and only one solution if and only if the so called 'Lipschitz conditions' are satisfied. One of these conditions requires that must have bounded partial derivatives in a 'small region' around and that is not for (1) if .

The fact is self evident if we try to solve (1), for example, with the condition . Proceeding in the 'standard way' you find that 'the solution' is...

(3)

All seems to be 'ok' but... but there is a little problem because it is not difficult to verify that this family of functions...

(4)

... where and is the so called 'step function' also satisfies the (1) with 'initial condition' ... just a little problem! (Headbang) ...

Kind regards