Challenging +1.2 This is a second-order vector differential equation with constant coefficients and exponential forcing term. While it requires knowledge of complementary function (SHM solution), particular integral (trying exponential form), and applying two initial conditions to find constants, the method is systematic and follows standard M5 procedures. The 11 marks reflect length rather than conceptual difficulty—it's more involved than average but doesn't require novel insight.
At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf{r}\) metres, where \(\mathbf{r}\) satisfies the vector differential equation
$$\frac{d^2\mathbf{r}}{dt^2} + 4\mathbf{r} = e^{2t} \mathbf{j}.$$
When \(t = 0\), \(P\) has position vector \((i + j)\) m and velocity \(2i\) m s\(^{-1}\).
Find an expression for \(\mathbf{r}\) in terms of \(t\).
[11]
At time $t$ seconds, the position vector of a particle $P$ is $\mathbf{r}$ metres, where $\mathbf{r}$ satisfies the vector differential equation
$$\frac{d^2\mathbf{r}}{dt^2} + 4\mathbf{r} = e^{2t} \mathbf{j}.$$
When $t = 0$, $P$ has position vector $(i + j)$ m and velocity $2i$ m s$^{-1}$.
Find an expression for $\mathbf{r}$ in terms of $t$.
[11]
\hfill \mbox{\textit{Edexcel M5 Q2 [11]}}