| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Implicit differential equation series solution |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring successive differentiation of an implicit differential equation to find higher derivatives, then constructing a Taylor series. While the technique is systematic, it requires careful algebraic manipulation and is beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07e Second derivative: as rate of change of gradient4.08a Maclaurin series: find series for function |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((1-x^2)\frac{d^3y}{dx^3} - 2x\frac{d^2y}{dx^2} - x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0\) | M1 | |
| At \(x = 0\), \(\frac{d^3y}{dx^3} = -\frac{dy}{dx} = -1\) | M1Acso3 | |
| (b) \(\left(\frac{d^2y}{dx^2}\right)_0 = -4\) | B1 | Allow anywhere |
| \(y = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3 + ...\) | ||
| \(= 2 - x - 2x^2 + \frac{1}{6}x^3 + ...\) | M1A1ft, A1 (dep)4 |
**(a)** $(1-x^2)\frac{d^3y}{dx^3} - 2x\frac{d^2y}{dx^2} - x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0$ | M1 |
At $x = 0$, $\frac{d^3y}{dx^3} = -\frac{dy}{dx} = -1$ | M1Acso3 |
**(b)** $\left(\frac{d^2y}{dx^2}\right)_0 = -4$ | B1 | Allow anywhere
$y = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3 + ...$ | |
$= 2 - x - 2x^2 + \frac{1}{6}x^3 + ...$ | M1A1ft, A1 (dep)4 |
**Total: [7]**
---10.
$$\left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$
At $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ at $x = 0$.
\item Express $y$ as a series in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.\\
(Total 7 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2007 Q10 [7]}}