Discussion
The initial value problem :
\[ y^{1} = f(x, y),\ y(x_{0}) = y_{0} \]
where \((x, y)\) belongs to a domain D, has a unique solution :
Answer: C
A unique solution is guaranteed if \(f(x, y)\) is continuous and satisfies a Lipschitz condition in D. (The paper writes \(y^{1}\) for the derivative \(y'\).)
This is the Picard-Lindelof theorem: continuity of \(f\) gives existence (Peano's theorem), and the Lipschitz condition \(|f(x, y_{1}) - f(x, y_{2})| \le L|y_{1} - y_{2}|\) makes the successive approximations converge to a single solution and rules out a second one.
Continuity alone, even with boundedness, gives existence but not uniqueness. The standard example is \(y' = 3y^{2/3}\), \(y(0) = 0\): \(f\) is continuous and bounded near the origin, yet both \(y = 0\) and \(y = x^{3}\) are solutions, because \(y^{2/3}\) is not Lipschitz at \(y = 0\). And not every function gives a unique solution, so 'for all functions' is false.
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