| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 This is a standard Further Maths FP2 question on second-order linear differential equations with constant coefficients. Part (a) requires finding the auxiliary equation (giving complex roots -1±2i), part (b) applies initial conditions to find constants in the exponentially damped oscillatory solution, and part (c) involves sketching a damped sinusoidal curve. While this is Further Maths content (making it harder than standard A-level), it follows a completely routine algorithmic procedure with no novel problem-solving required, placing it slightly above average difficulty overall. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10g Damped oscillations: model and interpret |
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
\item Given that $x = 1$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 1$ at $t = 0$, find the particular solution of the differential equation, giving your answer in the form $x = \mathrm { f } ( t )$.
\item Sketch the curve with equation $x = \mathrm { f } ( t ) , 0 \leq t \leq \pi$, showing the coordinates, as multiples of $\pi$, of the points where the curve cuts the $x$-axis.\\
(4)(Total 13 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2006 Q2 [13]}}