Question

When has initial condition y'=f(x,y), y(x_0)=y_0, x_0 and y_0 are known, at least one solution and when exactly one solution(sufficent condition)?

Answer 1

If f(x,y) is continuous on an open set around (x0,y0), then the IVP (initial value problem = ODE + initial condition) has a solution. That's from the Peano Theorem.

The Picard–Lindelöf theorem says that if f(x,y) is Lipschitz continuous, then the IVP has a solution and that solution is unique.

You should look both of them up so you get the hypotheses of the theorems precisely.