Problem

(a) Show that

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{2}x\sqrt{x^2-9}-\frac{9}{2}\cosh^{-1}\frac{x}{3}\right)=\sqrt{x^2-9}.$$

(b) Find the solution of the differential equation

$$x\frac{\mathrm{d}y}{\mathrm{d}x}-y=x^2\sqrt{x^2-9},$$

given that $y=1$ when $x=3$. Give your answer in the form $y=f(x)$.