| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - variable coefficients |
| Difficulty | Standard +0.3 This is a standard integrating factor question from Further Maths with straightforward identification of P(x) = tan x and Q(x) = cos³x. The integrating factor ∫tan x dx = ln|sec x| is a known result, and the integration ∫sec x · cos³x dx = ∫cos²x dx is routine using standard identities. The particular solution in part (ii) requires simple substitution. While it's Further Maths content, it follows a completely algorithmic procedure with no novel insight required, making it slightly easier than an average A-level question overall. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Integrating factor \(e^{\int \tan x(dx)}\) | B1 | For correct IF |
| \(= e^{-\ln\cos x}\) | M1 | For integrating to ln form |
| \(= (\cos x)^{-1}\) OR \(\sec x\) | A1 | For correct simplified IF AEF |
| \(\Rightarrow \frac{d}{dx}(y(\cos x)^{-1}) = \cos^2 x\) | B1√ | For \(\frac{d}{dx}(y \cdot \text{their IF}) = \cos^3 x \cdot \text{their IF}\) |
| \(y(\cos x)^{-1} = \int \frac{1}{2}(1+\cos 2x)(dx)\) | M1 | For integrating LHS |
| M1 | For attempting to use \(\cos 2x\) formula OR parts for \(\int \cos^2 x \, dx\) | |
| \(y(\cos x)^{-1} = \frac{1}{2}x + \frac{1}{4}\sin 2x + (c)\) | A1 | For correct integration both sides AEF |
| \(y = (\frac{1}{2}x + \frac{1}{4}\sin 2x + c)\cos x\) | A1 8 | For correct general solution AEF |
| (ii) \(2 = (\frac{1}{2}\pi + c) - 1 \Rightarrow c = 2 - \frac{1}{2}\pi\) | M1 | For substituting \((\pi, 2)\) into their GS and solve for \(c\) |
| \(y = (\frac{1}{2}x + \frac{1}{4}\sin 2x - 2 - \frac{1}{4}\pi)\cos x\) | A1 2 | For correct solution AEF |
**(i)** Integrating factor $e^{\int \tan x(dx)}$ | B1 | For correct IF
$= e^{-\ln\cos x}$ | M1 | For integrating to ln form
$= (\cos x)^{-1}$ OR $\sec x$ | A1 | For correct simplified IF AEF
$\Rightarrow \frac{d}{dx}(y(\cos x)^{-1}) = \cos^2 x$ | B1√ | For $\frac{d}{dx}(y \cdot \text{their IF}) = \cos^3 x \cdot \text{their IF}$
$y(\cos x)^{-1} = \int \frac{1}{2}(1+\cos 2x)(dx)$ | M1 | For integrating LHS
| M1 | For attempting to use $\cos 2x$ formula OR parts for $\int \cos^2 x \, dx$
$y(\cos x)^{-1} = \frac{1}{2}x + \frac{1}{4}\sin 2x + (c)$ | A1 | For correct integration both sides AEF
$y = (\frac{1}{2}x + \frac{1}{4}\sin 2x + c)\cos x$ | A1 8 | For correct general solution AEF
**(ii)** $2 = (\frac{1}{2}\pi + c) - 1 \Rightarrow c = 2 - \frac{1}{2}\pi$ | M1 | For substituting $(\pi, 2)$ into their GS and solve for $c$
$y = (\frac{1}{2}x + \frac{1}{4}\sin 2x - 2 - \frac{1}{4}\pi)\cos x$ | A1 2 | For correct solution AEF8 (i) Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$
expressing $y$ in terms of $x$ in your answer.\\
(ii) Find the particular solution for which $y = 2$ when $x = \pi$.
\hfill \mbox{\textit{OCR FP3 2007 Q8 [10]}}