2.
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$
- Show that
$$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$
where \(a\), \(b\) and \(c\) are integers to be determined.
Given that \(y = 1\) at \(x = 2\)
- determine the Taylor series expansion for \(y\) in ascending powers of \(( x - 2 )\), up to and including the term in \(( x - 2 ) ^ { 3 }\), giving each coefficient in simplest form.