| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.2 This is a second-order linear ODE with constant coefficients from FP3, requiring standard techniques: finding the complementary function (routine auxiliary equation), determining a particular integral (given the form, just differentiate and substitute), and applying initial conditions. While it involves resonance (forcing term matches CF frequency), the particular integral form is provided, removing the main conceptual challenge. The algebra is straightforward for Further Maths students, making this a solid but not exceptional FP3 question. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
The variables $x$ and $y$ satisfy the differential equation
$$\frac{\text{d}^2y}{\text{d}x^2} + 16y = 8\cos 4x.$$
\begin{enumerate}[label=(\roman*)]
\item Find the complementary function of the differential equation. [2]
\item Given that there is a particular integral of the form $y = px\sin 4x$, where $p$ is a constant, find the general solution of the equation. [6]
\item Find the solution of the equation for which $y = 2$ and $\frac{\text{d}y}{\text{d}x} = 0$ when $x = 0$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2010 Q6 [12]}}