It is given that \(v=y^{4}\) and
\(y^{3} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+3 y^{2}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+y^{3} \frac{\mathrm{~d} y}{\mathrm{~d} x}+y^{4}=\mathrm{e}^{-2 x} .\)
(a) Show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} v}{\mathrm{~d} x}+4 v=4 \mathrm{e}^{-2 x}\)
(b) Find \(y\) in terms of \(x\), given that, when \(x=0, y=1\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3}{8}\).