| Exam Board | Edexcel |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | First-order integration |
| Difficulty | Standard +0.8 This is a numerical methods question requiring application of finite difference approximations to a second-order nonlinear differential equation. While the technique is standard for FP1, students must carefully substitute the approximations, handle the nonlinear term y², rearrange to solve for y_{n+1}, and iterate twice. The algebraic manipulation and systematic application across multiple steps elevates this above routine FP1 questions but doesn't require deep insight. |
| Spec | 1.09g Numerical methods in context |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\left(\frac{dy}{dx}\right)_1 \approx \frac{(y_2-1)}{0.2}\) | B1 | Correct expression for first derivative using given values and approximation |
| \(\left(\frac{d^2y}{dx^2}\right)_1 \approx \frac{(y_2-2(2)+1)}{0.1^2}\) | B1 | Correct expression for second derivative using given values and approximation |
| \(\frac{(y_2-2(2)+1)}{0.1^2}+15\left(\frac{y_2-1}{0.2}\right)-3(2)^2=2(0.1) \Rightarrow y_2=...\) | M1 | Uses approximations, substitutes into ODE, obtains value for \(y\) at \(x=0.2\) |
| \(y_2 \approx \frac{1936}{875}\) (2.2125...) | A1 | Accept exact value or awrt 2.21 |
| \(\frac{(y_3-2\left(\frac{1936}{875}\right)+2)}{0.1^2}+15\left(\frac{y_3-2}{0.2}\right)-3\left(\frac{1936}{875}\right)^2=2(0.2) \Rightarrow y_3=...\) | M1 | Completes process using \(y\) at \(x=0.2\) to obtain \(y\) at \(x=0.3\) |
| \(y_3 \approx 2.32914...\) | A1 | Allow awrt 2.33 |
## Question 2:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\left(\frac{dy}{dx}\right)_1 \approx \frac{(y_2-1)}{0.2}$ | B1 | Correct expression for first derivative using given values and approximation |
| $\left(\frac{d^2y}{dx^2}\right)_1 \approx \frac{(y_2-2(2)+1)}{0.1^2}$ | B1 | Correct expression for second derivative using given values and approximation |
| $\frac{(y_2-2(2)+1)}{0.1^2}+15\left(\frac{y_2-1}{0.2}\right)-3(2)^2=2(0.1) \Rightarrow y_2=...$ | M1 | Uses approximations, substitutes into ODE, obtains value for $y$ at $x=0.2$ |
| $y_2 \approx \frac{1936}{875}$ (2.2125...) | A1 | Accept exact value or awrt 2.21 |
| $\frac{(y_3-2\left(\frac{1936}{875}\right)+2)}{0.1^2}+15\left(\frac{y_3-2}{0.2}\right)-3\left(\frac{1936}{875}\right)^2=2(0.2) \Rightarrow y_3=...$ | M1 | Completes process using $y$ at $x=0.2$ to obtain $y$ at $x=0.3$ |
| $y_3 \approx 2.32914...$ | A1 | Allow awrt 2.33 |
---\begin{enumerate}
\item The variables $x$ and $y$ satisfy the differential equation
\end{enumerate}
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 15 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 2 x$$
where $y = 1$ at $x = 0$ and where $y = 2$ at $x = 0.1$\\
Use the approximations
$$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$
with $h = 0.1$ to find an estimate for the value of $y$ when $x = 0.3$
\hfill \mbox{\textit{Edexcel FP1 AS 2021 Q2 [6]}}