Challenging +1.2 This is a standard second-order linear differential equation with constant coefficients and a particular integral requiring the method of undetermined coefficients. While it involves multiple steps (complementary function, particular integral, applying initial conditions, and asymptotic analysis), each step follows routine procedures taught in Further Pure 1. The asymptotic behavior analysis is straightforward since the complementary function contains exponentially decaying terms due to complex roots with negative real part. The question is slightly above average difficulty due to its length and the final asymptotic part, but requires no novel insight beyond standard FP1 techniques.
Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$
Given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\), find \(x\) in terms of \(t\).
Show that, for large positive values of \(t\) and for any initial conditions,
$$x \approx 5 \cos ( 2 t - \phi ) ,$$
where the constant \(\phi\) is such that \(\tan \phi = \frac { 4 } { 3 }\).
Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$
Given that $x = 5$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ when $t = 0$, find $x$ in terms of $t$.
Show that, for large positive values of $t$ and for any initial conditions,
$$x \approx 5 \cos ( 2 t - \phi ) ,$$
where the constant $\phi$ is such that $\tan \phi = \frac { 4 } { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2012 Q12 OR}}