2012 AJC MYE P1 Q5

Two variables \(x\) and \(y\) are related by the differential equation \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{2x}{\sqrt{1-4{{x}^{2}}}}\).

Given that \(y=1\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{2}\) when \(x=0\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=a\sqrt{1-4{{x}^{2}}}\) where \(a\) is a constant to be determined.[2]
2. Solve the differential equation in (i) by using the substitution \(2x=\sin \theta \), expressing the solution in the form \(2x=\sin \theta \), expressing the solution in the form \(y=\mathrm{f}\left( x \right)\).[6]