\begin{enumerate}
  \setcounter{enumi}{1}
  \item The vertical height, $h \mathrm {~m}$, above horizontal ground, of a passenger on a fairground ride, $t$ seconds after the ride starts, where $t \leqslant 5$, is modelled by the differential equation
\end{enumerate}

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} h } { \mathrm {~d} t } + 2 h = t ^ { 3 }$$

(a) Given that $t = \mathrm { e } ^ { x }$, show that\\
(i) $t \frac { \mathrm {~d} h } { \mathrm {~d} t } = \frac { \mathrm { d } h } { \mathrm {~d} x }$\\
(ii) $t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } h } { \mathrm {~d} x }$\\
(b) Hence show that the transformation $t = \mathrm { e } ^ { x }$ transforms equation (I) into the equation

$$\frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} h } { \mathrm {~d} x } + 2 h = \mathrm { e } ^ { 3 x }$$

(c) Hence show that

$$h = A t + B t ^ { 2 } + \frac { 1 } { 2 } t ^ { 3 }$$

where $A$ and $B$ are constants.

Given that when $t = 1 , h = 2.5$ and when $t = 2 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = - 1$\\
(d) determine the height of the passenger above the ground 5 seconds after the start of the ride.

\hfill \mbox{\textit{Edexcel FP1 2023 Q2}}