10.
$$\left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$
At \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1\).
- Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\).
- Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(Total 7 marks)