2023 EJC P2 Q3

The variables \(x\) and \(y\) are related by the differential equation \(\pi \frac{\mathrm{d}y}{\mathrm{d}x}+y\left( 3-\pi \tan x \right)=0\).

1. Using the substitution \(y=z\sec x\), show that \(\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{-3x}{\pi }\). Hence, show that the particular solution for which \(y=2\) when \(x=\frac{\pi }{3}\) can be expressed in the form \(y\cos x={{\mathrm{e}}^{a+bx}}\) where \(a\) and \(b\) are constants to be determined.[6]
2. For the graph of the solution found in part (a), find the equations of the two vertical asymptotes closest to the \(y\)-axis.[2]