| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.3 This is a Further Maths F2 question on resonance in second-order DEs. While the topic itself is advanced (requiring recognition that the forcing term matches the complementary function frequency), the question provides the form of the PI explicitly in part (a), reducing it to differentiation and substitution. Parts (b)-(d) are standard: finding general solution, applying initial conditions, and sketching. The conceptual leap (recognizing resonance requires modified PI) is bypassed by giving the form, making this a procedural multi-step question that's moderately challenging but not requiring deep insight. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| Differentiate twice: \(\frac{dy}{dx} = \lambda\sin 5x + 5\lambda x\cos 5x\) and \(\frac{d^2y}{dx^2} = 10\lambda\cos 5x - 25\lambda x\sin 5x\) | M1 A1 | |
| Substitute to give \(\lambda = \frac{3}{10}\) | M1 A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| Complementary function is \(y = A\cos 5x + B\sin 5x\) or \(Pe^{5ix} + Qe^{-5ix}\) | M1 A1 | |
| General solution: \(y = A\cos 5x + B\sin 5x + \frac{3}{10}x\sin 5x\) or exponential form | A1ft | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(y=0\) when \(x=0\) means \(A=0\) | B1 | |
| \(\frac{dy}{dx} = 5B\cos 5x + \frac{3}{10}\sin 5x + \frac{3}{2}x\cos 5x\) and at \(x=0\), \(\frac{dy}{dx}=5\) so \(5=5A\) | M1 M1 | |
| \(B = 1\) | A1 | |
| \(y = \sin 5x + \frac{3}{10}x\sin 5x\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| "Sinusoidal" through O with amplitude becoming larger | B1 | |
| Crosses x-axis at \(\frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}\) | B1 | (2) |
# Question 8:
## Part (a):
| Working/Answer | Marks | Notes |
|---|---|---|
| Differentiate twice: $\frac{dy}{dx} = \lambda\sin 5x + 5\lambda x\cos 5x$ and $\frac{d^2y}{dx^2} = 10\lambda\cos 5x - 25\lambda x\sin 5x$ | M1 A1 | |
| Substitute to give $\lambda = \frac{3}{10}$ | M1 A1 | (4) |
## Part (b):
| Working/Answer | Marks | Notes |
|---|---|---|
| Complementary function is $y = A\cos 5x + B\sin 5x$ or $Pe^{5ix} + Qe^{-5ix}$ | M1 A1 | |
| General solution: $y = A\cos 5x + B\sin 5x + \frac{3}{10}x\sin 5x$ or exponential form | A1ft | (3) |
## Part (c):
| Working/Answer | Marks | Notes |
|---|---|---|
| $y=0$ when $x=0$ means $A=0$ | B1 | |
| $\frac{dy}{dx} = 5B\cos 5x + \frac{3}{10}\sin 5x + \frac{3}{2}x\cos 5x$ and at $x=0$, $\frac{dy}{dx}=5$ so $5=5A$ | M1 M1 | |
| $B = 1$ | A1 | |
| $y = \sin 5x + \frac{3}{10}x\sin 5x$ | A1 | (5) |
## Part (d):
| Working/Answer | Marks | Notes |
|---|---|---|
| "Sinusoidal" through O with amplitude becoming larger | B1 | |
| Crosses x-axis at $\frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}$ | B1 | (2) |\begin{enumerate}
\item (a) Find the value of $\lambda$ for which $y = \lambda x \sin 5 x$ is a particular integral of the differential equation
\end{enumerate}
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
(b) Using your answer to part (a), find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
Given that at $x = 0 , y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 5$,\\
(c) find the particular solution of this differential equation, giving your solution in the form $y = \mathrm { f } ( x )$.\\
(d) Sketch the curve with equation $y = \mathrm { f } ( x )$ for $0 \leqslant x \leqslant \pi$.\\
\hfill \mbox{\textit{Edexcel F2 Q8 [14]}}