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.
A generic first order DE of the form...
... 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...