FP2 June 2012 Q4
4. Find the general solution of the differential equation \[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 5\frac{\mathrm{d}x}{\mathrm{d}t} + 6x = 2\cos t - \sin t\] (9)
| Scheme | Marks |
|---|---|
| \(m^2 + 5m + 6 = 0\) \(m = -2,\ -3\) | M1 |
| C.F. \((x =)A\mathrm{e}^{-2t} + B\mathrm{e}^{-3t}\) | A1 |
| P.I. \(x = P\cos t + Q\sin t\) | B1 |
| \(\dot{x} = -P\sin t + Q\cos t\) \(\ddot{x} = -P\cos t - Q\sin t\) | M1 |
| \((-P\cos t - Q\sin t) + 5(-P\sin t + Q\cos t) + 6(P\cos t + Q\sin t) = 2\cos t - \sin t\) | M1 |
| \(-P + 5Q + 6P = 2\) and \(-Q - 5P + 6Q = -1\), and solve for \(P\) and \(Q\) | M1 |
| \(P = \dfrac{3}{10}\) and \(Q = \dfrac{1}{10}\) | A1 A1 |
| \(x = A\mathrm{e}^{-2t} + B\mathrm{e}^{-3t} + \dfrac{3}{10}\cos t + \dfrac{1}{10}\sin t\) | B1 ft |
| (9) | |
| (9 marks) |
Notes
1st M1 form quadratic and attempt to solve (usual rules)
1st B1 Accept negative signs for coefficients. Coefficients must be different.
2nd M1 for differentiating their trig PI twice
3rd M1 for substituting \(x\), \(\dot{x}\) and \(\ddot{x}\) expressions
4th M1 Form 2 equations in two unknowns and attempt to solve
1st A1 for one correct, 2nd A1 for two correct
2nd B1 for \(x =\) their CF + their PI as functions of \(t\)
Condone use of the wrong variable (e.g. \(x\) instead of \(t\)) for all marks except final B1.