8.
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$
- Show that \(K t ^ { 2 } e ^ { 3 t }\) is a particular integral of the differential equation, where \(K\) is a constant to be found.
- 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 \(\boldsymbol { t } = \mathbf { 0 }\),
- 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
$$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
- 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.