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 interpreting the long-term behavior), each step follows routine procedures taught in Further Maths. The modeling context adds mild complexity but the mathematical techniques are all standard for FP1 level.
8 The value of the assets of a large commercial organisation at time \(t\), measured in years, is \(\\) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables \)y\( and \)t$ are related by the differential equation
$$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$
Find \(y\) in terms of \(t\), given that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2\) when \(t = 0\).
Show that, for large values of \(t\), the value of the assets is less than \(\\) 9.5 \times 10 ^ { 8 }$ for about a third of the time.
8 The value of the assets of a large commercial organisation at time $t$, measured in years, is $\$ \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)$. The variables $y$ and $t$ are related by the differential equation
$$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$
Find $y$ in terms of $t$, given that $y = 3$ and $\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2$ when $t = 0$.
Show that, for large values of $t$, the value of the assets is less than $\$ 9.5 \times 10 ^ { 8 }$ for about a third of the time.
\hfill \mbox{\textit{CAIE FP1 2002 Q8 [12]}}