| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.8 This is a standard second-order linear ODE with constant coefficients requiring auxiliary equation solution, particular integral, and applying initial conditions (8 marks suggests multi-step work). Part (b) requires finding and justifying a maximum using calculus. While methodical, it's routine Further Maths material with no novel insight required, placing it moderately above average difficulty. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral4.10f Simple harmonic motion: x'' = -omega^2 x |
$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$
Given that $x = 0$ and $\frac{dx}{dt} = 2$ at $t = 0$,
\begin{enumerate}[label=(\alph*)]
\item find $x$ in terms of $t$. [8]
\end{enumerate}
The solution to part (a) is used to represent the motion of a particle $P$ on the $x$-axis. At time $t$ seconds, where $t > 0$, $P$ is $x$ metres from the origin $O$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the maximum distance between $O$ and $P$ is $\frac{2\sqrt{3}}{9}$ m and justify that this distance is a maximum. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q8 [15]}}