- (a) Find the value of \(\lambda\) for which \(\lambda t ^ { 2 } \mathrm { e } ^ { 3 t }\) is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 6 \mathrm { e } ^ { 3 t } , \quad t \geqslant 0$$
(b) Hence find the general solution of this differential equation.
Given that when \(t = 0 , y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 4\)
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( t )\).