8 It is given that \(\mathbf { v } = y ^ { 4 }\) and
$$y ^ { 3 } \frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 y ^ { 2 } \left( \frac { d y } { d x } \right) ^ { 2 } + y ^ { 3 } \frac { d y } { d x } + y ^ { 4 } = e ^ { - 2 x }$$
- Show that
$$\frac { d ^ { 2 } v } { d x ^ { 2 } } + \frac { d v } { d x } + 4 v = 4 e ^ { - 2 x }$$
- Find \(y\) in terms of \(x\), given that, when \(x = 0 , y = 1\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 8 }\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.