FP2 January 2006 Q2

EdexcelOld spec13 marksSecond Order Differentials

(a) Find the general solution of the differential equation \[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0.\] (4)

(b) Given that \(x = 1\) and \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = 1\) at \(t = 0\), find the particular solution of the differential equation, giving your answer in the form \(x = \mathrm{f}(t)\). (5)

(c) Sketch the curve with equation \(x = \mathrm{f}(t)\), \(0 \leqslant t \leqslant \pi\), showing the coordinates, as multiples of \(\pi\), of the points where the curve cuts the \(x\)-axis. (4)

Mark scheme (a)

Scheme	Marks
\(m^2 + 2m + 5 = 0 \quad \Rightarrow \quad = -1 \pm 2\mathrm{i}\)	M1 A1
\(x = \mathrm{e}^{-t}(A\cos 2t + B\sin 2t)\) M: Correct form (needs the two different constants)	M1 A1
	(4)

Notes

First M: Form and attempt to solve auxiliary equation.

2^nd M: \(A\mathrm{e}^{(-1+2\mathrm{i})t} + 5\mathrm{e}^{(-1+2\mathrm{i})t}\) scores M1, as does \(A\mathrm{e}^{m_1t} + B\mathrm{e}^{m_2t}\) for real \(m_1, m_2\).

Confusion of variables: Can lose the final A mark in (a).

Mark scheme (b)

Scheme	Marks
\((1, 0) \quad \Rightarrow \quad A = 1\)	dB1
\(\dot{x} = -\mathrm{e}^{-t}(A\cos 2t + B\sin 2t) + \mathrm{e}^{-t}(-2A\sin 2t + 2B\cos 2t)\) M: Product diff. attempt	dM1
With \(A = 1\), \(\mathrm{e}^{-t}\{\cos 2t(-1 + 2B) + \sin 2t(-B - 2)\}\)
\(\dot{x} = 1, t = 0 \quad \Rightarrow \quad 1 = -A + 2B\)	M1
\(B = 1 \qquad (x = \mathrm{e}^{-t}(\cos 2t + \sin 2t))\) M: Use value of \(A\) to find \(B\).	dM1 A1cso
	(5)

Notes

B mark and first and third M marks are dependent on the M’s in part (a).

Sketch of x against t: starts at 1, rises to a maximum, crosses the t-axis at 3pi/8, dips to a minimum and crosses again at 7pi/8 — Mark scheme (c)

Scheme	Marks
‘Single oscillation’ between 0 and \(\pi\)	B1
Decreasing amplitude (dep. on a turning point)	B1ft
Initially increasing to maximum	B1ft
Any one correct intercept, whether in terms of \(\pi\) or not: 1 or \(\dfrac{3\pi}{8}\) or \(\dfrac{7\pi}{8}\) (Allow degrees: \(67.5^\circ\) or \(157.5^\circ\)) (Allow awrt \(0.32\pi\) or 1.18 or 2.75)	B1
	(4)
	(13 marks)