Challenging +1.8 This is a second-order vector differential equation requiring complementary function (solving auxiliary equation with roots ±2), particular integral (trying e^t form), and applying two initial conditions to find four constants. While methodical, it demands facility with vector DEs, exponential solutions, and careful algebraic manipulation—significantly above average A-level difficulty but standard for M5.
A particle \(P\) moves in the \(x\)-\(y\) plane so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation
$$\frac{d^2\mathbf{r}}{dt^2} - 4\mathbf{r} = -3e^t\mathbf{j}$$
When \(t = 0\), the particle is at the origin and is moving with velocity \((2\mathbf{i} + \mathbf{j})\) ms\(^{-1}\).
Find \(\mathbf{r}\) in terms of \(t\).
[10]
A particle $P$ moves in the $x$-$y$ plane so that its position vector $\mathbf{r}$ metres at time $t$ seconds satisfies the differential equation
$$\frac{d^2\mathbf{r}}{dt^2} - 4\mathbf{r} = -3e^t\mathbf{j}$$
When $t = 0$, the particle is at the origin and is moving with velocity $(2\mathbf{i} + \mathbf{j})$ ms$^{-1}$.
Find $\mathbf{r}$ in terms of $t$.
[10]
\hfill \mbox{\textit{Edexcel M5 2011 Q2 [10]}}