prove that

To prove:

$(2 x y)\left(\frac{d y}{d x}\right)=\left(\frac{x}{a}-\frac{a}{x}\right)$

Differentiating $y$ with respect to $x$

$\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)=\frac{1}{2 \sqrt{a x}}-\frac{\sqrt{a}}{2 x^{\frac{3}{2}}}$

Now,

$\mathrm{LHS}=(2 \mathrm{xy})\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

$L H S=2 x\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)\left(\frac{1}{2 \sqrt{a x}}-\frac{\sqrt{a}}{2 x^{\frac{3}{2}}}\right)$

$L H S=\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)\left(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}}\right)$

$L H S=\left(\frac{x}{a}-\frac{a}{x}\right)$

$\therefore \mathrm{LHS}=\mathrm{RHS}$