Solution: The given differential equation is
\[
\frac{\mathrm{d}y}{\mathrm{d}x} = \cos (xy).
\]
We will use
Picard-Lindelöf theorem to see if the given ODE has a (unique) solution or not. We consider the rectangle $R$ as
\[
R = \{ x: \vert x-0 \vert \leq a \} \times \{ y : \vert y - y_0 \vert \leq b \}
\]
for some $a$ and $b$. Let us denote $f(x,y(x)) = \cos (xy)$.
- It is clear that $f(x,y)$ is continuous in $x$.
- For checking the Lipschitz continuity of $f$ in $y$ we note that \begin{align*} \left\vert \frac{\partial f}{\partial y} \right\vert = \left\vert - x \sin (xy) \right\vert = \vert x \vert \leq a. \end{align*} Thus, $f$ is Lipschitz in $y$ with Lipschitz constant $a$. Thus, by using Picard-Lindelöf theorem, the given ODE has a unique solution in the rectangle $R$.
Therefore, the correct option will be
Option (A).