| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Standard +0.3 This is a straightforward integrating factor question from Further Maths. Part (a) requires verification by substitution (routine calculation), and part (b) involves standard application of the integrating factor method. While it's Further Maths content, the question follows a completely standard template with the integrating factor given, making it slightly easier than average overall. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| IF is \(e^{\int -\frac{2}{x}\,dx}\) | M1 | \(e^{\int \pm\frac{2}{x}\,dx}\) |
| \(= e^{-2\ln x}\) | A1 | P1 |
| \(= e^{\ln x^{-2}} = x^{-2} = \frac{1}{x^2}\) | A1 | Total: 3; AG — Be convinced |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\frac{d}{dx}\!\left(\frac{y}{x^2}\right) = \frac{1}{x^2}\,x\) | M1 | LHS as \(d/dx(y \times \text{IF})\) |
| A1 | PI | |
| \(\frac{y}{x^2} = \int \frac{1}{x}\,dx = \ln x + c\) | A1 | RHS; condone missing \(+c\) here |
| \(y = x^2 \ln x + cx^2\) | A1 | Total: 4 |
## Question 2:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|---------|
| IF is $e^{\int -\frac{2}{x}\,dx}$ | M1 | $e^{\int \pm\frac{2}{x}\,dx}$ |
| $= e^{-2\ln x}$ | A1 | P1 |
| $= e^{\ln x^{-2}} = x^{-2} = \frac{1}{x^2}$ | A1 | **Total: 3**; AG — Be convinced |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|---------|
| $\frac{d}{dx}\!\left(\frac{y}{x^2}\right) = \frac{1}{x^2}\,x$ | M1 | LHS as $d/dx(y \times \text{IF})$ |
| | A1 | PI |
| $\frac{y}{x^2} = \int \frac{1}{x}\,dx = \ln x + c$ | A1 | RHS; condone missing $+c$ here |
| $y = x^2 \ln x + cx^2$ | A1 | **Total: 4** |
---2
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { 1 } { x ^ { 2 } }$ is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = x$$
\item Hence find the general solution of this differential equation, giving your answer in the form $y = \mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2009 Q2 [7]}}