- The differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 8 \mathrm { e } ^ { - 3 t } \quad t \geqslant 0$$
describes the motion of a particle along the \(x\)-axis.
- Determine the general solution of this differential equation.
Given that the motion of the particle satisfies \(x = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\) when \(t = 0\)
- determine the particular solution for the motion of the particle.
On the graph of the particular solution found in part (b), the first turning point for \(t > 0\) occurs at \(x = a\).
- Determine, to 3 significant figures, the value of \(a\).
[0pt]
[Solutions relying entirely on calculator technology are not acceptable.]