Question: ivp has a unique solution?
Answer: Combining with existence thereom, the IVP (1) has unique solution y = y(x) defined in the interval |x − x0| ≤ α. |f(x, y1) − f(x, y2)| ≤ L|y1 − y2| ∀(x, yi) ∈ R. If ∂f/∂y exists and is bounded, then it necessarily saitisfies Lipschitz condition.