| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 second-order differential equation question with resonance. While it requires multiple techniques (finding particular integral with given form, complementary function, applying initial conditions, sketching), each step follows well-established procedures taught in FP2. The resonance case (auxiliary equation roots match forcing frequency) adds some complexity beyond basic C4 differential equations, but this is routine material for Further Maths students. The 14 marks reflect length rather than exceptional difficulty. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\alph*)]
\item Find the value of $z$ for which $y = zx \sin 5x$ is a particular integral of the differential equation
$$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [4]
\item Using your answer to part (a), find the general solution of the differential equation
$$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [3]
\end{enumerate}
Given that at $x = 0$, $y = 0$ and $\frac{dy}{dx} = 5$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the particular solution of this differential equation, giving your solution in the form $y = f(x)$. [5]
\item Sketch the curve with equation $y = f(x)$ for $0 \leq x \leq \pi$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q8 [14]}}