10. The following simultaneous equations are to be solved. $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 4 x + 2 y + 6 \mathrm { e } ^ { 3 t }
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 6 x + 8 y + 15 \mathrm { e } ^ { 3 t } \end{aligned}$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0\).
2. Given that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 9\) and \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 10\) when \(t = 0\), find the particular solution for \(x\) in terms of \(t\). Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}