7.
$$( 1 + 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = x + 4 y ^ { 2 }$$
- Show that
$$( 1 + 2 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 1 + 2 ( 4 y - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
- Differentiate equation 1 with respect to \(x\) to obtain an equation involving
$$\frac { \mathrm { d } ^ { 3 } } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x } , \quad x \text { and } y .$$
Given that \(y = \frac { 1 } { 2 }\) at \(x = 0\),
- find a series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(6)(Total 11 marks)