FP2 June 2015 Q3
3. Find, in the form \(y = \mathrm{f}(x)\), the general solution of the differential equation \[\tan x\,\frac{\mathrm{d}y}{\mathrm{d}x} + y = 3\cos 2x\tan x, \qquad 0 \lt x \lt \frac{\pi}{2}\] (6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} + \dfrac{y}{\tan x} = 3\cos 2x\) | |
| \(\displaystyle\int \cot x\,\mathrm{d}x = \ln|\sin x|\), IF \(= \sin x\) | M1 |
| \(\sin x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\cos x = 3\cos 2x\sin x\) | |
| \(y\sin x = \displaystyle\int 3\cos 2x\sin x\,\mathrm{d}x\) | M1A1 |
| \(y\sin x = \displaystyle\int 3\left(2\cos^2 x - 1\right)\sin x\,\mathrm{d}x\) or \(y\sin x = \dfrac{3}{2}\displaystyle\int\left(\sin 3x - \sin x\right)\mathrm{d}x\) | |
| \(y\sin x = 3\left[-\dfrac{2}{3}\cos^3 x + \cos x\right](+c)\) or \(y\sin x = \dfrac{3}{2}\left[-\dfrac{1}{3}\cos 3x + \cos x\right](+c)\) | dM1A1 |
| \(y = \dfrac{3\cos x - 2\cos^3 x + c'}{\sin x}\) oe or \(y = \dfrac{-3\cos 3x + 3\cos x + c'}{2\sin x}\) | B1ft (A1 on e-PEN) |
| (6 marks) |
Notes
M1: Divide by tan and attempt IF \(\mathrm{e}^{\int\cot x\,\mathrm{d}x}\) or equivalent needed
M1: Multiply through by IF and integrate LHS
A1: correct so far
dM1: dep (on previous M mark) integrate RHS using double angle or factor formula
\(k\cos^2 x\sin x \to \pm\cos^3 x,\ k\sin^2 x\cos x \to k\sin^3 x,\ \cos 3x \to \pm\dfrac{1}{3}\sin 3x,\ \sin 3x \to \pm\dfrac{1}{3}\cos 3x\)
A1: All correct so far constant not needed
B1ft: obtain answer in form \(y = \ldots\) any equivalent form Constant must be included and dealt with correctly. Award if correctly obtained from the previous line
Alternatives for integrating the RHS:
(i) By parts: Needs 2 applications of parts or one application followed by a trig method. Give M1 only if method is complete and A1 for a correct result.
(ii) \(y\sin x = \displaystyle\int 3\left(1 - 2\sin^2 x\right)\sin x\,\mathrm{d}x = \int 3\sin x - 6\sin^3 x\,\mathrm{d}x\)
Then use \(\sin 3x = 3\sin x - 4\sin^3 x\) to get \(y\sin x = \displaystyle\int \frac{3}{2}\left(\sin 3x - \sin x\right)\mathrm{d}x\) and integration shown above - both steps needed for M1
ALTERNATIVE: Mult through by \(\cos x\)
| Scheme | Marks |
|---|---|
| \(\sin x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\cos x = 3\cos 2x\sin x\) | M1 |
| \(y\sin x = \displaystyle\int 3\cos 2x\sin x\,\mathrm{d}x\) | M1A1 |
| Rest as main scheme |