2012 DHS P1 Q11(a) [Modified]

Given that \(z={{e}^{2x}}\frac{\mathrm{d}y}{\mathrm{d}x}\) , express \(\frac{\mathrm{d}z}{\mathrm{d}x}\) in terms of \(x\), \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}\).

Hence show that the differential equation \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\frac{\mathrm{d}y}{\mathrm{d}x}={{e}^{1-4x}}\)\(\left( * \right)\), can be reduced to the differential equation \(\frac{\mathrm{d}z}{\mathrm{d}x}={{\mathrm{e}}^{1-2x}}\).

Hence solve the differential equation \(\left( * \right)\), expressing \(y\) in terms of \(x\).