C4 June 2012 Q4
4. Given that \(y = 2\) at \(x = \dfrac{\pi}{4}\), solve the differential equation\[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3}{y\cos^2 x}\] (5)
| Scheme | Marks |
|---|---|
| \(\displaystyle\int y\,\mathrm{d}y = \int \frac{3}{\cos^2 x}\,\mathrm{d}x\) \(= \displaystyle\int 3\sec^2 x\,\mathrm{d}x\) Can be implied. Ignore integral signs | B1 |
| \(\dfrac{1}{2}y^2 = 3\tan x \quad (+C)\) | M1 A1 |
| \(y = 2,\ x = \dfrac{\pi}{4}\) \(\dfrac{1}{2}2^2 = 3\tan\dfrac{\pi}{4} + C\) | M1 |
| Leading to \(C = -1\) \(\dfrac{1}{2}y^2 = 3\tan x - 1\) or equivalent | A1 |
| (5) | |
| (5 marks) |
Notes
In the printed scheme a bracket joins these method marks: each later M mark is dependent on the M mark before it.