FP2 June 2010 Q2
2. The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation \[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + x + \cos x = 0\]
When \(t = 0\), \(x = 0\) and \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{1}{2}\).
Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t^3\). (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}'r'(t) = -x - \cos x,\qquad \mathrm{f}'r'(0) = -1\) | B1 |
| \(\mathrm{f}''r'(t) = (-1 + \sin x)\dfrac{\mathrm{d}x}{\mathrm{d}t},\qquad \mathrm{f}''r'(0) = -0.5\) | M1A1 |
| \(\mathrm{f}(t) = \mathrm{f}(0) + t\mathrm{f}r'(0) + \dfrac{t^2}{2}\mathrm{f}'r'(0) + \dfrac{t^3}{3!}\mathrm{f}''r'(0) + \ldots\) \(= 0.5t - 0.5t^2 - \frac{1}{12}t^3 + \ldots\) | M1 A1 |
| (5 marks) |