Challenging +1.2 This is a first-order linear vector differential equation requiring an integrating factor method applied component-wise. While M5 content is advanced, this is a standard textbook application: identify integrating factor e^t, multiply through, integrate both sides, and apply initial conditions. The technique is mechanical once recognized, requiring more algebraic care than conceptual insight.
A particle \(P\) moves in a plane so that its position vector, \(\mathbf{r}\) metres at time \(t\) seconds, satisfies the differential equation
$$\frac{d\mathbf{r}}{dt} + \mathbf{r} = t\mathbf{i} + e^{-t}\mathbf{j}$$
When \(t = 0\) 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 so that its position vector, $\mathbf{r}$ metres at time $t$ seconds, satisfies the differential equation
$$\frac{d\mathbf{r}}{dt} + \mathbf{r} = t\mathbf{i} + e^{-t}\mathbf{j}$$
When $t = 0$ 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 2014 Q2 [9]}}