11. The variable \(y\) satisfies the differential equation
$$4 \left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y$$
At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 }\).
- Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(x = 0\).
(1) (c) 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 \(x ^ { 3 }\).
- Find the value that the series gives for \(y\) at \(x = 0.1\), giving your answer to 5 decimal places.
(1)(Total 14 marks)