- The differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = \cos 3 t , \quad t \geqslant 0$$
describes the motion of a particle along the \(x\)-axis.
- Find the general solution of this differential equation.
- Find the particular solution of this differential equation for which, at \(t = 0\),
$$x = \frac { 1 } { 2 } \text { and } \frac { \mathrm { d } x } { \mathrm {~d} t } = 0$$
On the graph of the particular solution defined in part (b), the first turning point for \(t > 30\) is the point \(A\).
- Find approximate values for the coordinates of \(A\).