FP2 June 2014 (R) Q3
3.
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y\tan x = \mathrm{e}^{4x}\cos^2 x\) | |
| \(\mathrm{e}^{2\int \tan x\,\mathrm{d}x} = \mathrm{e}^{2\ln\sec x} = \sec^2 x\) or \(\dfrac{1}{\cos^2 x}\) | M1A1 |
| \(\sec^2 x\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y\tan x\sec^2 x = \mathrm{e}^{4x}\cos^2 x\sec^2 x\) | dM1 |
| \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(y\sec^2 x\right) = \mathrm{e}^{4x}\) | B1ft(\(y\sec^2 x\)) |
| \(y\sec^2 x = \dfrac{1}{4}\mathrm{e}^{4x} \quad (+c)\) | M1 |
| \(y = \left(\dfrac{1}{4}\mathrm{e}^{4x} + c\right)\cos^2 x\) oe | A1 |
| (6) |
Notes
M1 attempting the integrating factor, including integration of (2)\(\tan x\) \(\ln\cos\) or \(\ln\sec\) seen
A1 correct integrating factor \(\sec^2 x\) or \(\dfrac{1}{\cos^2 x}\)
M1 multiplying the equation by the integrating factor – may be implied by the next line.
B1ft \(y \times\) their IF
M1 attempting a complete integration of rhs Must include \(k\mathrm{e}^{4x}\) but \(4\mathrm{e}^{4x}\) would imply differentiation. Constant not needed (Incorrect IF may lead to integration by parts, so integration must be complete)
A1 correct solution in form \(y = \ldots\) constant must be included
| Scheme | Marks |
|---|---|
| \(y = 1,\ \ x = 0 \quad 1 = \left(\dfrac{1}{4} + c\right)\) | M1 |
| \(c = \dfrac{3}{4}\) | |
| \(y = \dfrac{1}{4}\left(\mathrm{e}^{4x} + 3\right)\cos^2 x\) oe | A1 |
| (2) | |
| (8 marks) |
Notes
M1 using given initial conditions to obtain a value for \(c\)
A1 fully correct final answer May be in the form \(y\sec^2 x = \ldots\) or \(4y\sec^2 x = \ldots\)