Challenging +1.3 This is a first-order linear vector differential equation requiring an integrating factor method. While M5 content is advanced, this is a standard application of the integrating factor technique (μ = t^(-2)) followed by integration and applying initial conditions. The vector nature adds minimal complexity since components can be treated independently. More routine than typical M5 questions involving complex multi-body systems or variable mass problems.
A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds \((t > 0)\) satisfies the differential equation
$$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4\mathbf{i}$$
When \(t = 1\), the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m.
Find \(\mathbf{r}\) in terms of \(t\).
[9]
A particle $P$ moves in a plane such that its position vector $\mathbf{r}$ metres at time $t$ seconds $(t > 0)$ satisfies the differential equation
$$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4\mathbf{i}$$
When $t = 1$, the particle is at the point with position vector $(\mathbf{i} + \mathbf{j})$ m.
Find $\mathbf{r}$ in terms of $t$.
[9]
\hfill \mbox{\textit{Edexcel M5 2012 Q1 [9]}}