3.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = x - y ^ { 2 }$$
- Show that
$$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a y \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + c \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) ^ { 2 }$$
where \(a\), \(b\) and \(c\) are integers to be determined.
- Hence find a series solution, in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\), of the differential equation (I), given that \(y = 1\) at \(x = 0\)