| 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 and a non-homogeneous term, requiring complementary function (repeated root case), particular integral (trying x = A cos 3t + B sin 3t), and applying initial conditions. Part (c) adds a numerical/graphical element requiring understanding of long-term behavior. While methodical, it's more demanding than typical A-level pure maths due to the algebraic manipulation and being Further Maths content, but follows standard FP2 procedures without requiring novel insight. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
\includegraphics{figure_1}
The differential equation
$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 9x = \cos 3t, \quad t \geq 0,$$
describes the motion of a particle along the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of this differential equation. [8]
\item Find the particular solution of this differential equation for which, at $t = 0$, $x = \frac{1}{2}$ and $\frac{dx}{dt} = 0$. [5]
\end{enumerate}
On the graph of the particular solution defined in part (b), the first turning point for $T > 30$ is the point $A$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find approximate values for the coordinates of $A$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q6 [15]}}