Standard +0.3 This is a straightforward application of the integrating factor method to a vector differential equation. While it involves vectors, the technique is standard: find integrating factor e^(2t), multiply through, integrate, and apply initial conditions. The exponential integration is routine and the vector nature doesn't add significant complexity since components can be handled separately. Slightly above average difficulty due to the vector context and being from M5, but still a textbook exercise requiring no novel insight.
2. At time \(t\) seconds the position vector of a particle \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation
$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$
Given that \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j }\) when \(t = 0\), find \(\mathbf { r }\) in terms of \(t\).
(Total 7 marks)
2. At time $t$ seconds the position vector of a particle $P$, relative to a fixed origin $O$, is $\mathbf { r }$ metres, where $\mathbf { r }$ satisfies the differential equation
$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$
Given that $\mathbf { r } = 2 \mathbf { i } - \mathbf { j }$ when $t = 0$, find $\mathbf { r }$ in terms of $t$.\\
(Total 7 marks)\\
\hfill \mbox{\textit{Edexcel M5 2005 Q2 [7]}}