# Applying Cauchys theorem

• Dec 2nd 2009, 01:38 PM
georgiahelo
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!!
• Dec 3rd 2009, 01:49 AM
chisigma
A generic first order DE of the form...

$\displaystyle y^{'} = f(x,y)$ (1)

... with 'initial condition' $\displaystyle y(x_{0})= y_{0}$ does admit one and only one solution if the so called 'Cauchy-Lipschitz conditions' are satisfied. In particular $\displaystyle f(*,*)$ must be continuous and with bounded partial derivative respect to y in a 'small region' around $\displaystyle [x_{0},y_{0}]$. In this case is $\displaystyle f(x,y) = \frac{1}{y}$ and that means that any 'initial condition' with $\displaystyle y_{0}\ne 0$ satisfies the conditions to have only and only one solution...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Dec 3rd 2009, 07:29 AM
Pedro²
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