Q: Given that $\displaystyle y = \mathrm{arcsin}\left( \frac{x}{a} \right)$, where $\displaystyle a$ is a constant, show that $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - x \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right) ^3 = 0 $.

My method:

$\displaystyle y = \mathrm{arcsin}\left( \frac{x}{a} \right)$

$\displaystyle \sin y = \frac{1}{a} . x$

$\displaystyle \cos y \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{a}$

$\displaystyle - \sin y \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right) ^2 + \cos y \left( \frac{\mathrm{d}^2y}{\mathrm{d}x^2} \right) = 0$

But that isn't what they require. What do I do? Thanks in advance for the help.