Standard +0.8 This is a standard Further Maths second-order non-homogeneous differential equation requiring both complementary function (solving auxiliary equation with complex roots) and particular integral (using undetermined coefficients with trigonometric RHS). While methodical, it demands multiple techniques and careful algebra, placing it moderately above average difficulty.
1 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = \sin x$$
It must be of form \(y =\) "their CF+PI" and of form "\(a\cos x + b\sin x\) with \(a\) or \(b\) nonzero plus standard CF form" with 2 constants and not in complex exponential form
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$\lambda = -2 \pm 3i$ | M1 | ft on complex $\lambda$ only
$e^{-2x}(A\cos 3x + B\sin 3x)$ | A1ft | condone $Ae^{(-2\pm3i)x} + Be^{(-2\mp3i)x}$
$y' = -a\sin x + b\cos x$, $y'' = -a\cos x - b\sin x$ in DE: $-a\cos x - b\sin x + 4(-a\sin x + b\cos x) + 13(a\cos x + b\sin x) = \sin x$ | M1 | Differentiate twice and substitute
$12a + 4b = 0$, $12b - 4a = 1$ | M1 | Compare
$a = -\frac{1}{40}$, $b = \frac{3}{40}$ | A1 |
$y = \frac{1}{40}(3\sin x - \cos x) + e^{-2x}(A\cos 3x + B\sin 3x)$ | A1ft | It must be of form $y =$ "their CF+PI" and of form "$a\cos x + b\sin x$ with $a$ or $b$ nonzero plus standard CF form" with 2 constants and not in complex exponential form
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