2020 NYJC CT2 P1 Q10

The motion of a particle is described by the differential equation \(\frac{{{\mathrm{d}}^{2}}x}{\mathrm{d}{{t}^{2}}}-4\frac{\mathrm{d}x}{\mathrm{d}t}+4x={{\mathrm{e}}^{2t}}\), where \(x\) is the displacement of the particle in metres along the \(x\)- axis with respect to origin \(O\) at time \(t\) seconds. By using the substitution \(x=u{{\mathrm{e}}^{2t}}\), show that this differential equation may be reduced to the form \(\frac{{{\mathrm{d}}^{2}}u}{\mathrm{d}{{t}^{2}}}=a\), where \(a\) is a constant to be determined.

Hence, find the displacement of the particle at time \(t\) if the particle moves off from the initial point \(x=1\) with a velocity of \(1\)m/s.[7]

After some time, the particle is returned to its initial point and set into a new motion along the \(x\)- axis. The motion of the particle is described by the differential equation \(\frac{\mathrm{d}v}{\mathrm{d}t}=2{{\mathrm{e}}^{-v}}\), where \(v\) is the velocity of the particle in metres per second at time \(t\) seconds after it starts on this new motion. If the particle is at rest just before it begins this new motion, find an expression for \(t\) in terms of \(v\).

Show that \(2x=3+{{\mathrm{e}}^{v}}(v-1)\), where \(x\) is the horizontal displacement of the particle in metres.[6]