| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - variable coefficients |
| Difficulty | Standard +0.3 This is a standard first-order linear differential equation requiring the integrating factor method. While it's Further Maths content, the technique is routine: identify P(x)=x, find integrating factor e^(x²/2), multiply through, integrate, and apply initial condition. The algebra is straightforward with no tricks, making it slightly easier than average overall but typical for FP3. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Integrating factor: \(e^{\int \text{d}x} = e^x\) | B1 | For correct IF |
| \(\frac{d}{dx}(ye^{x^2}) = xe^{x^2}\) | M1 | For \(\frac{d}{dx}(\text{y·their IF}) = xe^{\frac{1}{2}x^2}\) ·their IF |
| \(ye^{\frac{1}{2}x^2} = \frac{1}{2}e^{x^2} (+c)\) | A1 | For correct integration both sides |
| \(y = e^{-\frac{1}{2}x^2}(\frac{1}{2}e^{x^2} + c) = \frac{1}{2}e^{\frac{1}{2}x^2} + ce^{-\frac{1}{2}x^2}\) | A1 4 | For correct solution AEF as \(y = f(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((0, 1) \Rightarrow c = \frac{1}{2}\) | M1 | For substituting \((0,1)\) into their GS, solving for \(c\) and obtaining a solution of the DE |
| \(y = \frac{1}{2}(e^{\frac{1}{2}x^2} + e^{-\frac{1}{2}x^2})\) | A1 2 | For correct solution AEF (Allow \(y = \cosh(\frac{1}{2}x^2)\)) |
## (i)
Integrating factor: $e^{\int \text{d}x} = e^x$ | B1 | For correct IF
$\frac{d}{dx}(ye^{x^2}) = xe^{x^2}$ | M1 | For $\frac{d}{dx}(\text{y·their IF}) = xe^{\frac{1}{2}x^2}$ ·their IF
$ye^{\frac{1}{2}x^2} = \frac{1}{2}e^{x^2} (+c)$ | A1 | For correct integration both sides
$y = e^{-\frac{1}{2}x^2}(\frac{1}{2}e^{x^2} + c) = \frac{1}{2}e^{\frac{1}{2}x^2} + ce^{-\frac{1}{2}x^2}$ | A1 4 | For correct solution **AEF** as $y = f(x)$
## (ii)
$(0, 1) \Rightarrow c = \frac{1}{2}$ | M1 | For substituting $(0,1)$ into their GS, solving for $c$ and obtaining a solution of the DE
$y = \frac{1}{2}(e^{\frac{1}{2}x^2} + e^{-\frac{1}{2}x^2})$ | A1 2 | For correct solution **AEF** (Allow $y = \cosh(\frac{1}{2}x^2)$)
---\begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac{dy}{dx} + xy = xe^{\frac{x^2}{2}},$$
giving your answer in the form $y = f(x)$. [4]
\item Find the particular solution for which $y = 1$ when $x = 0$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2011 Q1 [6]}}