| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Explicit differential equation series solution |
| Difficulty | Challenging +1.2 This is a standard FP2 Taylor series question requiring differentiation of the given DE to find higher derivatives, then substitution of the initial condition to build the series term-by-term. Part (a) is routine implicit differentiation, and part (b) follows a well-practiced algorithm. While it requires careful algebraic manipulation and is harder than typical C3/C4 content due to being Further Maths, it's a textbook exercise without novel problem-solving demands. |
| Spec | 1.07q Product and quotient rules: differentiation4.08a Maclaurin series: find series for function |
5.
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x + y ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 - 2 y ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 3$$
Given that $y = 1$ at $x = 1$,
\item find a series solution for $y$ in ascending powers of ( $x - 1$ ), up to and including the term in $( x - 1 ) ^ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2012 Q5 [10]}}