## (i) METHOD 1

$m + 4(0) \Rightarrow$ CF $(y) = Ae^{-4x}$ | M1 A1 2 | For correct auxiliary equation (soi); For correct CF

## METHOD 2

Separating variables on $\frac{dy}{dx} + 4y = 0$ |  |

$\Rightarrow \ln y = -4x$ | M1 | For integration to this stage

$\Rightarrow$ CF $(y) = Ae^{-4x}$ | A1 | For correct CF

## (ii)

PI $(y) = p\cos 3x + q\sin 3x$ | B1 | For stating PI of correct form

$y' = -3p\sin 3x + 3q\cos 3x$ | M1 | For substituting $y$ and $y'$ into DE

$\Rightarrow (-3p+4q)\sin 3x + (4p+3q)\cos 3x = 5\cos 3x$ | A1 | For correct equation

$\Rightarrow \begin{cases} -3p+4q = 0 \\ 4p+3q = 5 \end{cases} \Rightarrow p = \frac{4}{5}, q = \frac{3}{5}$ | M1 A1 A1 | For equating coeffs and solving; For correct value of $p$, and of $q$

GS $(y) = Ae^{-4x} + \frac{4}{5}\cos 3x + \frac{3}{5}\sin 3x$ | B1 \/ 7 | For GS f.t. from their CF+PI with 1 arbitrary constant in CF and none in PI

SR Integrating factor method may be used, followed by 2-stage integration by parts or C+iS method
Marks for (i) are awarded only if CF is clearly identified

## (iii)

$e^{-4x} \to 0, \frac{4}{5}\cos 3x + \frac{3}{5}\sin 3x = \frac{\sin(3x+\alpha)}{\cos(3x+\alpha)}$ | M1 | For considering either term

$\Rightarrow -1 \leq y \leq 1$ OR $-1 \lesssim y \lesssim 1$ | A1 \/ 2 | For correct range (allow $<$) CWO f.t. as $-\sqrt{p^2+q^2} \leq y \leq \sqrt{p^2+q^2}$ from (ii)

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