Thread: Applying 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!!

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

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