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Let's address each statement one by one: ### i) The initial value problem: \[ \frac{dy}{dx} = x^2 + y^2, \quad y(0) = 0 \] This statement asks whether there exists a unique solution to this initial value problem in some interval. Answer: False. The differential equation \(\frac{dy}{dx} = x^2 + y^2\) does not satisfy the conditions for the existence and uniqueness of a solution according to the Picard-Lindelöf theorem (which is based on the Lipschitz continuity of the function with respect to \(y\)). Here, \(f(x, y) = x^2 + y^2\) is not Lipschitz continuous in \(y\) because it grows quadratically in \(y\), which violates the condition for uniqueness. Therefore, this problem may have multiple solutions (or ______ ___ ____ _________ _____ _______ _________.
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