2.
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { x } \left( 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right)$$
- Show that
$$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \mathrm { e } ^ { x } \left[ 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + k y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right]$$
where \(k\) is a constant to be found.
Given that, at \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\),
- find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).