| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Challenging +1.2 This is a standard second-order linear differential equation with constant coefficients from Further Maths. Part (i) requires routine application of complementary function and particular integral methods. Part (ii) adds mild complexity by requiring interpretation of asymptotic behavior to determine arbitrary constants, but the technique is straightforward once the general solution is found. The question is harder than typical A-level pure maths but represents a standard FP3 exercise rather than requiring novel insight. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(m^2+4m=0 \Rightarrow m=0,-4\) | M1 | For attempt to solve correct auxiliary equation |
| \(\text{CF} = A+Be^{-4x}\) | A1 | For correct CF |
| PI \(y=pe^{2x} \Rightarrow 4p+8p=12\) | B1 | For PI of correct form seen. Beware poor use of \(pxe^{2x}\); scores maximum M1 A1 B0 M1 A0 B0 |
| Differentiating PI and substituting | M1 | |
| \(\Rightarrow p=1\) | A1 | For correct \(p\) |
| GS \(y = A+Be^{-4x}+e^{2x}\) | B1FT | For using GS = CF + PI with 2 arbitrary constants in GS and none in PI |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrating \(\Rightarrow \frac{dy}{dx}+4y = 6e^{2x}+c\) | M1 | For attempt to integrate equation |
| \(+c\) included | B1 | |
| IF \(e^{4x} \Rightarrow \frac{d}{dx}(ye^{4x}) = 6e^{6x}+ce^{4x}\) | B1\(\sqrt{}\) | For correct IF, f.t. from their DE |
| M1 | For multiplying through by their IF and attempting to integrate | |
| \(\Rightarrow ye^{4x} = e^{6x}+\frac{1}{4}ce^{4x}+B\) | A1 | For correct integration both sides, including \(+B\) |
| \(\Rightarrow y = e^{2x}+A+Be^{-4x}\) | A1 | For correct solution. Must include "\(y=\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = -4Be^{-4x}+2e^{2x}\) | M1 | For differentiating "their GS" with 2 arbitrary constants and substituting values to obtain an equation. If "their CF" is \((A+Bx)e^{-4x}\) can score max of M1 A0 B1 A0 |
| \(\left(0,\frac{dy}{dx}=6\right)\Rightarrow -4B+2=6 \Rightarrow B=-1\) | A1 | For correct \(B\) |
| \((y \approx e^{2x})\Rightarrow A=0\) | B1 | For correct \(A\) and consistent with "their GS" |
| \(\Rightarrow y = -e^{-4x}+e^{2x}\) | A1 | For correct equation www |
# Question 6(i):
**Method 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2+4m=0 \Rightarrow m=0,-4$ | M1 | For attempt to solve correct auxiliary equation |
| $\text{CF} = A+Be^{-4x}$ | A1 | For correct CF |
| PI $y=pe^{2x} \Rightarrow 4p+8p=12$ | B1 | For PI of correct form seen. Beware poor use of $pxe^{2x}$; scores maximum M1 A1 B0 M1 A0 B0 |
| Differentiating PI and substituting | M1 | |
| $\Rightarrow p=1$ | A1 | For correct $p$ |
| GS $y = A+Be^{-4x}+e^{2x}$ | B1FT | For using GS = CF + PI with 2 arbitrary constants in GS and none in PI |
**Method 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrating $\Rightarrow \frac{dy}{dx}+4y = 6e^{2x}+c$ | M1 | For attempt to integrate equation |
| $+c$ included | B1 | |
| IF $e^{4x} \Rightarrow \frac{d}{dx}(ye^{4x}) = 6e^{6x}+ce^{4x}$ | B1$\sqrt{}$ | For correct IF, f.t. from their DE |
| | M1 | For multiplying through by their IF and attempting to integrate |
| $\Rightarrow ye^{4x} = e^{6x}+\frac{1}{4}ce^{4x}+B$ | A1 | For correct integration both sides, including $+B$ |
| $\Rightarrow y = e^{2x}+A+Be^{-4x}$ | A1 | For correct solution. Must include "$y=$" |
---
# Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = -4Be^{-4x}+2e^{2x}$ | M1 | For differentiating "their GS" with 2 arbitrary constants and substituting values to obtain an equation. If "their CF" is $(A+Bx)e^{-4x}$ can score max of M1 A0 B1 A0 |
| $\left(0,\frac{dy}{dx}=6\right)\Rightarrow -4B+2=6 \Rightarrow B=-1$ | A1 | For correct $B$ |
| $(y \approx e^{2x})\Rightarrow A=0$ | B1 | For correct $A$ and consistent with "their GS" |
| $\Rightarrow y = -e^{-4x}+e^{2x}$ | A1 | For correct equation www |
---6 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 12 \mathrm { e } ^ { 2 x }$$
(i) Find the general solution of the differential equation.\\
(ii) It is given that the curve which represents a particular solution of the differential equation has gradient 6 when $x = 0$, and approximates to $y = \mathrm { e } ^ { 2 x }$ when $x$ is large and positive. Find the equation of the curve.
\hfill \mbox{\textit{OCR FP3 2012 Q6 [10]}}