Challenging +1.2 This is a standard Further Maths FP2 Taylor series question requiring systematic differentiation of the differential equation to find successive derivatives at t=0, then constructing the series. While it involves multiple steps and careful algebra with the cos x term, the method is algorithmic and well-practiced in FP2. The 5-mark allocation and straightforward structure place it moderately above average difficulty.
The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation
$$\frac{d^2 x}{dt^2} + x + \cos x = 0.$$
When \(t = 0\), \(x = 0\) and \(\frac{dx}{dt} = \frac{1}{2}\).
Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t^4\). [5]
The displacement $x$ metres of a particle at time $t$ seconds is given by the differential equation
$$\frac{d^2 x}{dt^2} + x + \cos x = 0.$$
When $t = 0$, $x = 0$ and $\frac{dx}{dt} = \frac{1}{2}$.
Find a Taylor series solution for $x$ in ascending powers of $t$, up to and including the term in $t^4$. [5]
\hfill \mbox{\textit{Edexcel FP2 Q2 [5]}}