Reducing Differential Equations (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

The motion of a particle $P$ along the $x$-axis is modelled by the differential equation

$t^2\frac{{\text{d}}^{2}x}{\text{d}{t}^2}-t(3t+2)\frac{\text{d}x}{\text{d}t}+(2t^2+3t+2)x=12t^3{\text{e}}^{3t}\left(\text{I}\right)$

where $P$ has displacement $x$ metres from the origin $O$ at time $t$ minutes, $t>0$.

Show that the transformation $x=tu$ transforms the differential equation ($\mathrm{I}$) into the differential equation

$\frac{{\text{d}}^{2}u}{\text{d}{t}^2}-3\frac{\text{d}u}{\text{d}t}+2u=12{\text{e}}^{3t}$

Given that $P$ is at $O$ when $t=\ln 2$ and when $t=\ln 3$, determine the particular solution of the differential equation (I).