Discussion
The initial value problem
\[ \begin{aligned} &\frac{dy}{dx} = \cos(xy),\ x \in \mathbb{R}, \\ &y(0) = y_{0}, \end{aligned} \]
where \(y_{0}\) is a real constant, has
Answer: A
The problem has a unique solution.
Let f(x, y) = cos(xy). It is continuous, and its partial derivative \(\partial f/\partial y = -x\sin(xy)\) is bounded by |x|, so f is Lipschitz in y on every strip |x| ≤ a. By the Picard-Lindelof theorem a unique local solution exists through (0, \(y_{0}\)).
Since |f| ≤ 1, the solution cannot blow up in finite x, and the strip-wise Lipschitz bound lets the unique solution be continued over all of \(\mathbb{R}\).
Two or infinitely many solutions arise only when uniqueness fails, as for \(y' = \sqrt{|y|}\), where the right side is not Lipschitz. 'No solution' is ruled out by Peano's existence theorem, since f is continuous.
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