M5 January 2006 Q3
3. The position vector \(\mathbf{r}\) of a particle \(P\) at time \(t\) satisfies the vector differential equation
\[\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} + 2\mathbf{r} = 4\mathbf{i}.\]Given that the position vector of \(P\) at time \(t = 0\) is \(2\mathbf{j}\), find the position vector of \(P\) at time \(t\).
| Scheme | Marks |
|---|---|
| Either CF: \(\dot{\mathbf{r}} + 2\mathbf{r} = \mathbf{0}\) | |
| \(\Rightarrow\ \mathbf{r} = \mathbf{A}\mathrm{e}^{-2t}\) | M1 A1 |
| PI \(\mathbf{r} = 2\mathbf{i}\) | B1 |
| GS \(\mathbf{r} = \mathbf{A}\mathrm{e}^{-2t} + 2\mathbf{i}\) | A1 |
| \(t = 0\), \(\mathbf{r} = 2\mathbf{j} \;\Rightarrow\; \mathbf{A} = 2\mathbf{j} - 2\mathbf{i}\) | M1 |
| \(\mathbf{r} = (2\mathbf{j} - 2\mathbf{i})\mathrm{e}^{-2t} + 2\mathbf{i}\) or \(2\mathbf{i}(1 - \mathrm{e}^{-2t}) + 2\mathbf{j}\mathrm{e}^{-2t}\) | A1 |
| (6 marks) |
Notes
Or
| IF \(= \mathrm{e}^{2t}\) | B1 |
| \(\dfrac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\mathrm{e}^{2t}) = 4\mathbf{i}\mathrm{e}^{2t}\) | M1 |
| \(\mathbf{r}\mathrm{e}^{2t} = 2\mathbf{i}\mathrm{e}^{2t} + \mathbf{A}\) | A1 |
| \(\mathbf{r} = 2\mathbf{i} + \mathbf{A}\mathrm{e}^{-2t}\) | A1 |
then as above (M1 A1).