The curve \(C\) has equation
$$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
- Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\). - Show that \(A _ { 1 } = 0\).
- Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\). - Show that for \(n \geqslant 2\),
$$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
- Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
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