10.
$$y \frac { d ^ { 2 } y } { d x ^ { 2 } } + \left( \frac { d y } { d x } \right) ^ { 2 } + y = 0$$
- Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\),
- find the series solution for \(y\), in ascending powers of \(x\), up to an including the term in \(x ^ { 3 }\).
- Comment on whether it would be sensible to use your series solution to give estimates for \(y\) at \(x = 0.2\) and at \(x = 50\).