| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Implicit differential equation series solution |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring implicit differentiation of a nonlinear differential equation to find higher derivatives, then constructing a Taylor series. The implicit differentiation with product rule applied multiple times is algebraically demanding, and students must carefully substitute initial conditions. However, it follows a standard FP2 template for Taylor series from differential equations, making it harder than typical A-level but not requiring exceptional insight. |
| Spec | 4.08a Maclaurin series: find series for function |
$$y\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + y = 0.$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac{d^3 y}{dx^3}$. [5]
Given that $y = 1$ and $\frac{dy}{dx} = 1$ at $x = 0$,
\item find the series solution for $y$, in ascending powers of $x$, up to an including the term in $x^3$. [5]
\item Comment on whether it would be sensible to use your series solution to give estimates for $y$ at $x = 0.2$ and at $x = 50$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q10 [12]}}