apply the existence and uniqueness theorem of Cauchy to the equation

dy/dx = 1/y

with the boundary condition

y(0) = y

to decide for what values of y the solution may or may not exist and be unique.

im confused!!

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- Dec 2nd 2009, 01:38 PMgeorgiaheloApplying Cauchys theorem
apply the existence and uniqueness theorem of Cauchy to the equation

dy/dx = 1/y

with the boundary condition

y(0) = y

to decide for what values of y the solution may or may not exist and be unique.

im confused!! - Dec 3rd 2009, 01:49 AMchisigma
A generic first order DE of the form...

(1)

... with 'initial condition' does admit one and only one solution if the so called 'Cauchy-Lipschitz conditions' are satisfied. In particular must be continuous and with bounded partial derivative respect to y in a 'small region' around . In this case is and that means that any 'initial condition' with satisfies the conditions to have only and only one solution...

Kind regards

- Dec 3rd 2009, 07:29 AMPedro²
In fact, you can relax the continuity of the partial derivative respecto to y (it isn't necessary that f be differentiable). If f satisfy a local Lipschitz condition in the variable y, the statement of the theorem also holds.

Cya :D