CAIE A-Level Further Math A2 2.6.1 Linear Differential Equations Questions

$\mathrm{e}^{\int 3 \mathrm{~d} x}=\mathrm{e}^{3 x}$
Finds integrating factor.

$\frac{\mathrm{d}}{\mathrm{d} x}\left(y \mathrm{e}^{3 x}\right)=\mathrm{e}^{3 x} \sin x$
Correct form on LHS and attempt
to integrate RHS.

EITHER
$$\begin{aligned} \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x=-\mathrm{e}^{3 x} \cos x+3 \int \mathrm{e}^{3 x} \cos x \mathrm{~d} x \text { OR } \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x=\frac{1}{3} \mathrm{e}^{3 x} \sin x-\frac{1}{3} \int \mathrm{e}^{3 x} \cos x \mathrm{~d} x \end{aligned}$$
Integrates by parts once or uses
$$\sin x=\frac{e^{\mathrm{i} x}-\mathrm{e}^{-\mathrm{i} x}}{2 \mathrm{i}}.$$

EITHER
$$\begin{aligned} \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x=-\mathrm{e}^{3 x} \cos x+3\left(\mathrm{e}^{3 x} \sin x-3 \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x\right) \text { OR } \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x=\frac{1}{3} \mathrm{e}^{3 x} \sin x-\frac{1}{3}\left(\frac{1}{3} \mathrm{e}^{3 x} \cos x+\frac{1}{3} \int \mathrm{e}^{3 x} \sin x \mathrm{~d} x\right) \end{aligned}$$
Integrates by parts again or substitutes $\frac{\mathrm{e}^{\mathrm{i} x}-\mathrm{e}^{-\mathrm{i} x}}{2 \mathrm{i}}=\sin x$ and
$$\frac{\mathrm{e}^{\mathrm{i} x}+\mathrm{e}^{-\mathrm{i} x}}{2}=\cos x.$$$y \mathrm{e}^{3 x}=\frac{1}{10} \mathrm{e}^{3 x}(3 \sin x-\cos x)+C$
Must not see i.

$1=-\frac{1}{10}+C$
Finds C. Substitutes into their
expression (must be integrated).

$y=\frac{3}{10} \sin x-\frac{1}{10} \cos x+\frac{11}{10} e^{-3 x}$
Divides through by coefficient of y.

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