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...
$\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
$\displaystyle \chi$ $\displaystyle \sigma$