Challenging +1.2 This is a standard FP2 resonance case question where the PI form is given, requiring differentiation, substitution, and coefficient matching to find λ, then combining with complementary function and applying initial conditions. While it involves multiple steps and the resonance concept (modified PI with x factor), the structure is highly procedural and follows textbook methods exactly. The sketch in part (d) is routine for this topic. Slightly above average difficulty due to being Further Maths content and requiring careful algebraic manipulation, but this is a classic exam question type that well-prepared FP2 students would recognize immediately.
8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
(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 \(0 \leqslant x \leqslant \pi\).
Complementary function is \(y = A\cos 5x + B\sin 5x\) or \(Pe^{5ix} + Qe^{-5ix}\)
M1 A1
General solution is \(y = A\cos 5x + B\sin 5x + \frac{3}{10}x\sin 5x\) (or exponential form)
A1ft
(3 marks total)
Part (c):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\(y=0\) 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
Wait — \(5=5B\)
\(B = 1\)
A1
\(y = \sin 5x + \frac{3}{10}x\sin 5x\)
A1
(5 marks total)
Part (d):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
"Sinusoidal" through \(O\), 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 marks total)
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Could you share the actual mark scheme pages? They would typically contain question numbers, model answers, mark allocations (M1, A1, B1, etc.), and examiner guidance notes. I'd be happy to extract and format that content once you provide those pages.
## Question 8:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = \lambda\sin 5x + 5\lambda x\cos 5x$ 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 marks total) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Complementary function is $y = A\cos 5x + B\sin 5x$ or $Pe^{5ix} + Qe^{-5ix}$ | M1 A1 | |
| General solution is $y = A\cos 5x + B\sin 5x + \frac{3}{10}x\sin 5x$ (or exponential form) | A1ft | (3 marks total) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=0$ 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 | Wait — $5=5B$ |
| $B = 1$ | A1 | |
| $y = \sin 5x + \frac{3}{10}x\sin 5x$ | A1 | (5 marks total) |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| "Sinusoidal" through $O$, 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 marks total) |
The image you've shared appears to be just a back/copyright page from an Edexcel publication (Summer 2010), containing only publisher contact information and registration details. There is **no mark scheme content** visible on this page.
Could you share the actual mark scheme pages? They would typically contain question numbers, model answers, mark allocations (M1, A1, B1, etc.), and examiner guidance notes. I'd be happy to extract and format that content once you provide those pages.
8. (a) Find the value of $\lambda$ for which $y = \lambda x \sin 5 x$ is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
(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 FP2 2010 Q8 [14]}}