5.
$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 0$$
Given that at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
- show that, at \(x = 0 , \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 3 } { 2 }\)
- Find a series solution for \(y\) up to and including the term in \(x ^ { 3 }\)