It is given that $w = \cos y$ and

$$\tan y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + 2 \tan y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + \mathrm { e } ^ { - 2 x } \sec y$$

(i) Show that

$$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = - \mathrm { e } ^ { - 2 x }$$

(ii) Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = \frac { 1 } { 3 } \pi$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 3 } }$. [10]\\

\hfill \mbox{\textit{CAIE FP1 2019 Q11 EITHER [10]}}