| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Standard non-homogeneous with trigonometric RHS |
| Difficulty | Challenging +1.8 This FP3 second-order differential equation question requires finding the complementary function (routine), verifying a given particular integral through differentiation and algebraic manipulation (technically demanding with multiple steps), and determining validity domains. The algebraic verification in part (ii)(a) involving trigonometric identities and the need to recognize where cosec x is undefined elevates this above standard exercises, though the particular integral form is provided rather than requiring students to construct it themselves. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08i Integration by parts4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\roman*)]
\item Find the complementary function of the differential equation
$$\frac{d^2y}{dx^2} + y = \cosec x.$$ [2]
\item It is given that $y = p(\ln \sin x) \sin x + qx \cos x$, where $p$ and $q$ are constants, is a particular integral of this differential equation.
\begin{enumerate}[label=(\alph*)]
\item Show that $p - 2(p + q) \sin^2 x \equiv 1$. [6]
\item Deduce the values of $p$ and $q$. [2]
\end{enumerate}
\item Write down the general solution of the differential equation. State the set of values of $x$, in the interval $0 \leqslant x \leqslant 2\pi$, for which the solution is valid, justifying your answer. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q8 [13]}}