| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Find turning points or extrema |
| Difficulty | Standard +0.3 This is a standard FP2 second-order differential equation question with auxiliary equation method, particular integral (polynomial form), and application of initial conditions. Part (c) requires finding a minimum using calculus, but all techniques are routine for Further Maths students. The question is methodical rather than requiring insight, making it slightly easier than average for A-level overall but typical for FP2. |
| Spec | 4.10b Model with differential equations: kinematics and other contexts4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation
$$2\frac{d^2 x}{dt^2} + 5\frac{dx}{dt} + 2x = 2t + 9.$$ [6]
\item Find the particular solution of this differential equation for which $x = 3$ and $\frac{dx}{dt} = -1$ when $t = 0$. [4]
The particular solution in part (b) is used to model the motion of a particle $P$ on the $x$-axis. At time $t$ seconds ($t \geq 0$), $P$ is $x$ metres from the origin $O$.
\item Show that the minimum distance between $O$ and $P$ is $\frac{1}{2}(5 + \ln 2)$ m and justify that the distance is a minimum. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q44 [14]}}