Challenging +1.8 This is a second-order vector differential equation requiring particular integral and complementary function methods. While M5 students are trained in these techniques, the question demands careful handling of exponential solutions, vector components, and applying two initial conditions to find four constants. The systematic approach is standard but the execution requires precision across multiple steps, making it significantly harder than average A-level questions but within expected M5 scope.
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 \((2i + j)\) 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 $(2i + j)$ ms$^{-1}$.
Find $\mathbf{r}$ in terms of $t$.
[10]
\hfill \mbox{\textit{Edexcel M5 Q2 [10]}}