Verify that implicitly defined function satisfies differential equation

Problem 1. on page 9 in George Simmons' textbook "Differential equations with applications and historical notes" reads "verify that the following functions (explicit or implicit) are solutions of the corresponding differential equations" and further (h):
$$y=\sin^{-1}xy\quad\quad xy'+y=y'\sqrt{1-x^2y^2}$$
So $y$ is implicitly defined. Could you provide me with an approach/Ansatz to this?

Answer

If you use implicit differentiation i.e assume $y = y(x)$ then,

$$\begin{align*} \frac{d}{dx} (y) &= \frac{d}{dx} (\textrm{arcsin}(xy)) \end{align*}$$