Find the particular solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$
given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
Show that, for large positive values of \(t\) and for any initial conditions,
$$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$
where the constant \(\phi\) is such that \(\tan \phi = 6\).
10 (i) Find the particular solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$
given that $x = 3$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ when $t = 0$.\\
(ii) Show that, for large positive values of $t$ and for any initial conditions,
$$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$
where the constant $\phi$ is such that $\tan \phi = 6$.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q10}}