Challenging +1.2 This is a standard second-order linear differential equation with constant coefficients and a particular integral involving trigonometric functions. While it requires multiple techniques (auxiliary equation, particular integral by undetermined coefficients, and understanding that complementary function terms decay for large x), these are all routine Further Maths procedures. The asymptotic behavior follows directly from recognizing exponential decay, making this a straightforward multi-step question slightly above average difficulty due to the Further Maths context and multiple parts.
8 Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$
Show that, for large positive \(x\) and independently of the initial conditions,
$$y \approx R \sin ( 3 x + \phi )$$
where the constants \(R\) and \(\phi\), such that \(R > 0\) and \(0 < \phi < 2 \pi\), are to be determined correct to 2 decimal places.
8 Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$
Show that, for large positive $x$ and independently of the initial conditions,
$$y \approx R \sin ( 3 x + \phi )$$
where the constants $R$ and $\phi$, such that $R > 0$ and $0 < \phi < 2 \pi$, are to be determined correct to 2 decimal places.
\hfill \mbox{\textit{CAIE FP1 2010 Q8 [9]}}