| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Differential equation given |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring repeated differentiation of an implicit differential equation using product and chain rules, then constructing a Taylor series. Part (a) demands careful algebraic manipulation through multiple derivatives, while part (b) requires systematic evaluation at x=0 and series construction. The complexity of the expressions and multi-step nature elevates this above standard A-level, though it follows a predictable Taylor series methodology for FP2. |
| Spec | 4.08a Maclaurin series: find series for function |
$$\frac{d^2 y}{dx^2} = e^x \left(2x \frac{dy}{dx} + y^2 + 1\right).$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{d^4 y}{dx^4} = e^x \left[2x \frac{d^3 y}{dx^3} + 4 \frac{d^2 y}{dx^2} + 6y \frac{dy}{dx} + y^2 + 1\right],$$
where $k$ is a constant to be found. [3]
\end{enumerate}
Given that, at $x = 0$, $y = 1$ and $\frac{dy}{dx} = 2$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find a series solution for $y$ in ascending powers of $x$, up to and including the term in $x^4$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q2 [7]}}