8.

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$
\begin{enumerate}[label=(\alph*)]
\item Show that $K t ^ { 2 } e ^ { 3 t }$ is a particular integral of the differential equation, where $K$ is a constant to be found.
\item Find the general solution of the differential equation. (3)

Given that a particular solution satisfies $\boldsymbol { y } = 3$ and $\frac { \mathrm { d } y } { \mathrm {~d} t } = 1$ when $\boldsymbol { t } = \mathbf { 0 }$,
\item find this solution.(4)

Another particular solution which satisfies $\boldsymbol { y } = \mathbf { 1 }$ and $\frac { \mathrm { d } y } { \mathrm {~d} t } = \mathbf { 0 }$ when $\boldsymbol { t } = \mathbf { 0 }$, has equation

$$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
\item For this particular solution draw a sketch graph of $y$ against $t$, showing where the graph crosses the $t$-axis. Determine also the coordinates of the minimum of the point on the sketch graph.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FP2 2003 Q8 [16]}}