Standard +0.8 This is a standard Further Maths second-order linear ODE with constant coefficients and trigonometric forcing term. Students must find the complementary function (requiring complex roots from the auxiliary equation) and a particular integral (using undetermined coefficients with both sin x and cos x terms). While methodical, it requires multiple techniques and careful algebra, placing it moderately above average difficulty but still a routine FP3 question.
5 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 85 \cos x .$$
Condone \(Ae^{(-1+3i)x} + Be^{(-1-3i)x}\); ft on complex \(\lambda\) only
PI: \(y = a\cos x + b\sin x\)
B1
Trial function \(y = a\cos x\) scores max B0 M1 M0
\(y' = -a\sin x + b\cos x\), \(y'' = -a\cos x - b\sin x\); sub into DE: \(-a\cos x - b\sin x + 2(-a\sin x + b\cos x) + 10(a\cos x + b\sin x) = 85\cos x\)
M1*
Differentiate twice and substitute
\(-a + 2b + 10a = 85\); \(-b - 2a + 10b = 0\)
M1*
Compare coefficients
\(a = 9, b = 2\)
A1
PI correct
GS: \(y = 9\cos x + 2\sin x + e^{-x}(A\cos 3x + B\sin 3x)\)
*A1ft
Their CF (standard form) + their PI; dep on both M1 marks
# Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| AE: $\lambda^2 + 2\lambda + 10 = 0$ | B1 | |
| $\lambda = -1 \pm 3i$ | B1 | |
| CF: $e^{-x}(A\cos 3x + B\sin 3x)$ | B1ft | Condone $Ae^{(-1+3i)x} + Be^{(-1-3i)x}$; ft on complex $\lambda$ only |
| PI: $y = a\cos x + b\sin x$ | B1 | Trial function $y = a\cos x$ scores max B0 M1 M0 |
| $y' = -a\sin x + b\cos x$, $y'' = -a\cos x - b\sin x$; sub into DE: $-a\cos x - b\sin x + 2(-a\sin x + b\cos x) + 10(a\cos x + b\sin x) = 85\cos x$ | M1* | Differentiate twice and substitute |
| $-a + 2b + 10a = 85$; $-b - 2a + 10b = 0$ | M1* | Compare coefficients |
| $a = 9, b = 2$ | A1 | PI correct |
| GS: $y = 9\cos x + 2\sin x + e^{-x}(A\cos 3x + B\sin 3x)$ | *A1ft | Their CF (standard form) + their PI; dep on both M1 marks |
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