Let's analyze each statement step by step with explanations, then conclude whether they are True or False. --- **Statement (a):** *There is a solution but uniqueness can only be guaranteed for some α.* **Analysis:** - The problem describes an initial value problem (IVP): \[ y'(t) = f(t, y(t)), \quad y(0) = y_0 \] - The function \(f(t, y)\) is continuous and uniformly Lipschitz in \(y\), which (by the Picard-Lindelöf theorem) guarantees **existence and uniqueness** of solutions **locally** (in some interval around \(t=0\)). - The statement suggests that **solution exists**, but **uniqueness** can only be guaranteed **for some \(\alpha\)**. Since the Lipschitz condition is uniform in \(t\) over \([0, T]\), and the problem states this holds **for all** \(t \in [0, T]\), the **uniqueness** is guaranteed **globally** on \([0, T]\), not just for some \(\alpha\). - Therefore, the statement's implication that uniqueness is only guaranteed for some \(\alpha\) (possibly a small interval) **is not** accurate when the Lipschitz condition holds uniformly on the entire interval. **Conclusion:** *The statement is False.* **Because** with the uniform Lipschitz condition on \([0, T]\), **uniqueness is guaranteed on the entire interval**, not just some \(\alpha\). --- **Statement (b):** *Euler's method is a multistep method to find approximation of the IVP's solution.* **Analysis:** - **Euler's method** is **a single-step method**, not a multistep method. - **Multistep methods** (like Adams-Bashforth, Adams-Moulton) use multiple previous points to compute the next value. - **Euler's method** uses only the current point to compute the next, so it is **not** multistep. **Conclusion:** *The statement is False.* **Because** Euler's method is a **single-step** method, not multistep. --- **Statement (c):** *The local discretization error is a measure of how well the solution \( y(t) \) of the IVP satisfies the numerical method formula locally.* **Analysis:** - **Local discretization error** (also called local truncation error) measures **how accurately** the numerical method approximates the exact solution **over a single step**, assuming the previous values are exact. - It quantifies **the difference** between the exact solution's behavior and what the numerical method produces over that one step. - So, **it is a measure of how well the exact solution satisfies** the numerical scheme **locally**. **Conclusion:** *The statement is True.* **Because** the local discretization error indeed measures **how well the exact solution satisfies** the numerical method **locally**. --- ### **Final summary:** | Statement | True/False | Explanation | |------------|--------------|-------------| | (a) | False | Uniqueness is guaranteed globally on \([0, T]\) under uniform Lipschitz conditions. | | (b) | False | Euler's method is a single-step method, not multistep. | | (c) | True | The local discretization error measures how well the exact solution satisfies the numerical scheme locally. | --- **Overall conclusion:** - Statements (a) and (b) are **False** - Statement (c) is **True**