| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.8 This is a resonance problem in second-order DEs requiring the modified particular integral form (multiplying by x) when the forcing frequency matches the natural frequency. While the technique is standard for FP3, it requires recognizing resonance, correctly applying the modified PI method, and analyzing long-term behavior—going beyond routine calculation to require conceptual understanding of why resonance causes unbounded growth versus bounded oscillation in the non-resonant case. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| PI: \(y=ax\cos 2x+bx\sin 2x\) | ||
| \(\frac{dy}{dx}=a\cos 2x-2ax\sin 2x+b\sin 2x+2bx\cos 2x\) | B1 | For correct \(\frac{dy}{dx}\) or better |
| \(\frac{d^2y}{dx^2}=-4a\sin 2x-4ax\cos 2x+4b\cos 2x-4bx\sin 2x\) | ||
| substituting into DE and comparing coefficients: \(-4a=1, 4b=0\) | M1, M1 | Differentiate twice and substitute; compare coefficients |
| \(\Rightarrow a=-\frac{1}{4}, b=0\) | A1 | |
| AE: \(\lambda^2+4=0\), \(\lambda=\pm 2i\) | M1 | For correct auxiliary equation and attempt to solve |
| CF: \(A\cos 2x+B\sin 2x\) | A1 | oe form |
| GS: \(y=\left(A-\frac{1}{4}x\right)\cos 2x+B\sin 2x\) | A1ft | Must be real and contain 2 unknowns |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| oscillations | B1 | oe (accept sketch) dep consistent with 6(i) |
| unbounded | B1 | oe; if zero, sc1 for recognition that \(x\cos 2x\) term becomes dominant |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| If \(k\neq 2\) then PI \(y=\alpha\cos kx+\beta\sin kx\) | B1 | |
| So bounded oscillations | B1 | oe (accept sketch) |
# Question 6:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| PI: $y=ax\cos 2x+bx\sin 2x$ | | |
| $\frac{dy}{dx}=a\cos 2x-2ax\sin 2x+b\sin 2x+2bx\cos 2x$ | B1 | For correct $\frac{dy}{dx}$ or better |
| $\frac{d^2y}{dx^2}=-4a\sin 2x-4ax\cos 2x+4b\cos 2x-4bx\sin 2x$ | | |
| substituting into DE and comparing coefficients: $-4a=1, 4b=0$ | M1, M1 | Differentiate twice and substitute; compare coefficients |
| $\Rightarrow a=-\frac{1}{4}, b=0$ | A1 | |
| AE: $\lambda^2+4=0$, $\lambda=\pm 2i$ | M1 | For correct auxiliary equation and attempt to solve |
| CF: $A\cos 2x+B\sin 2x$ | A1 | oe form |
| GS: $y=\left(A-\frac{1}{4}x\right)\cos 2x+B\sin 2x$ | A1ft | Must be real and contain 2 unknowns |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| oscillations | B1 | oe (accept sketch) dep consistent with 6(i) |
| unbounded | B1 | oe; if zero, sc1 for recognition that $x\cos 2x$ term becomes dominant |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| If $k\neq 2$ then PI $y=\alpha\cos kx+\beta\sin kx$ | B1 | |
| So bounded oscillations | B1 | oe (accept sketch) |
---6 The differential equation $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin k x$ is to be solved, where $k$ is a constant.\\
(i) In the case $k = 2$, by using a particular integral of the form $a x \cos 2 x + b x \sin 2 x$, find the general solution.\\
(ii) Describe briefly the behaviour of $y$ when $x \rightarrow \infty$.\\
(iii) In the case $k \neq 2$, explain whether $y$ would exhibit the same behaviour as in part (ii) when $x \rightarrow \infty$.
\hfill \mbox{\textit{OCR FP3 2013 Q6 [11]}}