(a) Given that \(x = t ^ { \frac { 1 } { 2 } }\), determine, in terms of \(y\) and \(t\),
\(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
\(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
(b) Hence show that the transformation \(x = t ^ { \frac { 1 } { 2 } }\), where \(t > 0\), transforms the differential equation
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 6 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 9 x ^ { 3 } y = x ^ { 5 }$$
into the differential equation
$$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = t$$
(c) Solve differential equation (II) to determine a general solution for \(y\) in terms of \(t\).
(d) Hence determine the general solution of differential equation (I).