- The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t ( t + 1 ) \frac { \mathrm { d } x } { \mathrm {~d} t } + 2 ( t + 1 ) x = 8 t ^ { 3 } \mathrm { e } ^ { t }$$
where \(P\) has displacement \(x\) metres from the origin \(O\) at time \(t\) minutes, \(t > 0\)
- Show that the transformation \(x = t u\) transforms the differential equation (I) into the differential equation
$$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} u } { \mathrm {~d} t } = 8 \mathrm { e } ^ { t }$$
Given that \(P\) is at \(O\) when \(t = \ln 3\) and when \(t = \ln 5\)
- determine the particular solution of the differential equation (I)