FP2 June 2006 Q7
7. \[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 3\sin x = 0. \qquad \text{At } t = 0,\ \ x = 0 \ \text{ and } \ \frac{\mathrm{d}x}{\mathrm{d}t} = 0.4\]
(b) Find a series solution for \(x\), in ascending powers of \(t\), up to and including the term in \(t^3\). (4)
(c) Use your answer to (b) to obtain an estimate of \(x\) at \(t = 0.3\). (2)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}''(t) = -3\sin x, \qquad \mathrm{f}''(0) = 0\) | |
| \(\mathrm{f}'''(t) = -3\cos x\,\dfrac{\mathrm{d}x}{\mathrm{d}t}, \qquad \mathrm{f}'''(0) = -3 \times 0.4 = -1.2\) | M1 A1 |
| \(\mathrm{f}(t) = \mathrm{f}(0) + \mathrm{f}'(0) + \dfrac{t^2}{2}\mathrm{f}''(0) + \dfrac{t^3}{3!}\mathrm{f}'''(0) + \ldots\) | |
| \(= 0.4t - 0.2t^3\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Substituting \(t = 0.3\) into their answer to (b) and evaluating | M1 |
| \(\mathrm{f}(0.3) \approx 0.1146\) cao | A1 |
| (2) |