| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Iterative/numerical methods |
| Difficulty | Standard +0.3 This is a straightforward application of given numerical approximation formulas (Euler's method and midpoint method) with explicit instructions. Students simply substitute values into provided formulas and perform basic arithmetic—no derivation, conceptual understanding of numerical methods, or problem-solving required. Slightly above trivial due to careful arithmetic with decimals, but easier than average A-level questions. |
| Spec | 1.09e Iterative method failure: convergence conditions4.10a General/particular solutions: of differential equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{y_1 - 0.2}{0.1} \approx \left(\frac{dy}{dx}\right)_0 = 0.2 \times e^0(= 0.2)\) | M1 | |
| \(y_1 \approx 0.22\) | A12 | |
| (b) \(\left(\frac{dy}{dx}\right)_{0.2} \approx 0.22 \times e^{0.01} \approx 0.2222...\) | B1 | |
| \(\frac{0.2222...}{0.2} \approx 0.2222...\) | M1 | |
| \(y_2 \approx 0.2444\) | cao | A13 |
**(a)** $\frac{y_1 - 0.2}{0.1} \approx \left(\frac{dy}{dx}\right)_0 = 0.2 \times e^0(= 0.2)$ | M1 |
$y_1 \approx 0.22$ | A12 |
**(b)** $\left(\frac{dy}{dx}\right)_{0.2} \approx 0.22 \times e^{0.01} \approx 0.2222...$ | B1 |
$\frac{0.2222...}{0.2} \approx 0.2222...$ | M1 |
$y_2 \approx 0.2444$ | cao | A13 |
**Total: [5]**
---9.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \mathrm { e } ^ { x ^ { 2 } } .$$
It is given that $y = 0.2$ at $x = 0$.
\begin{enumerate}[label=(\alph*)]
\item Use the approximation $\frac { y _ { 1 } - y _ { 0 } } { h } \approx \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) _ { 0 }$, with $h = 0.1$, to obtain an estimate of the value of $y$ at $x = 0.1$.
\item Use your answer to part (a) and the approximation $\frac { y _ { 2 } - y _ { 0 } } { 2 h } \approx \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) _ { 1 }$, with $h = 0.1$, to obtain an estimate of the value of $y$ at $x = 0.2$.
Gives your answer to 4 decimal places.\\
(Total 5 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2007 Q9 [5]}}