Problem

(a) Show that

$$\frac{\mathrm{d}}{\mathrm{d}x} \bigg( \frac{1}{2}x\sqrt{4 - x^2} + 2\sin^{-1}\bigg(\frac{1}{2}x\bigg) \bigg) = \sqrt{4 - x^2}.$$

(b) Find the solution of the differential equation

$$2\frac{\mathrm{d}y}{\mathrm{d}x} + \frac{y}{2 + x} = 2\sqrt{2 - x}$$

for which $y = \frac{1}{2}$ when $x = 1$. Give your answer in an exact form.