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