M5 June 2013 Q1
1. Solve the differential equation
\[\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} - 2\mathbf{r} = \mathbf{0}\]given that when \(t = 0\), \(\mathbf{r}.\mathbf{j} = 0\) and \(\mathbf{r} \times \mathbf{j} = \mathbf{i} + \mathbf{k}\). (7)
| Scheme | Marks |
|---|---|
| G.S. is \(\mathbf{r} = \mathbf{A}\mathrm{e}^{2t}\) | B1 |
| \(t = 0 : \mathbf{A}.\mathbf{j} = 0 \Rightarrow \mathbf{A} = p\mathbf{i} + r\mathbf{k}\) | M1 A1 |
| \((p\mathbf{i} + r\mathbf{k}) \times \mathbf{j} = \mathbf{i} + \mathbf{k}\) | M1 A1 |
| \(-r\mathbf{i} + p\mathbf{k} = \mathbf{i} + \mathbf{k} \Rightarrow r = -1;\ p = 1\) | M1 |
| \(\mathbf{r} = (\mathbf{i} - \mathbf{k})\mathrm{e}^{2t}\) | A1 |
| (7 marks) |
Notes
B1 for \(\mathbf{r} = \mathbf{A}\mathrm{e}^{2t}\) oe.
First M1 for use of initial conditions \(t = 0\), \(\mathbf{r}.\mathbf{j} = 0\). (M0 if no explicit \(\mathbf{r}\) expression to sub into)
First A1 for \(\mathbf{A} = p\mathbf{i} + r\mathbf{k}\) (or \(q = 0\))
Second M1 for attempt at cross-product \(\mathbf{r} \times \mathbf{j}\) when \(t = 0\) (M0 if no explicit \(\mathbf{r}\) expression to sub into)
Second A1 for \((-r\mathbf{i} + p\mathbf{k})\)
Third M1 for using the second condition to find values for \(p\) and \(r\).
Third A1 for a correct answer.
N.B. All marks available apart from the final A1 if unsound work seen e.g. logs of vectors, provided that logs are removed at the start to give an explicit expression for \(\mathbf{r}\) which can be evaluated in order to find the value of the constant.