| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| 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 straightforward application of the integrating factor method with standard steps: verifying the given integrating factor (routine calculation), multiplying through, integrating (requiring substitution u = x³+1), and applying initial conditions. While it requires multiple techniques, each step follows a well-established procedure with no novel insight needed, making it slightly easier than average for Further Maths. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| IF is \(\exp\left(\int \dfrac{2}{x}\,dx\right)\) | M1 | And with integration attempted |
| \(= e^{2\ln x}\) | A1 | |
| \(= x^2\) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\dfrac{d}{dx}\left[yx^2\right] = 3x^2(x^3+1)^{\frac{1}{2}}\) | M1A1 | PI |
| \(\Rightarrow yx^2 = \dfrac{2}{3}(x^3+1)^{\frac{3}{2}} + A\) | m1 | \(k(x^3+1)^{\frac{3}{2}}\) |
| A1 | Condone missing '\(A\)' | |
| \(\Rightarrow 4 = \dfrac{2}{3}(9)^{\frac{3}{2}} + A\) | m1 | Use of boundary conditions to find constant |
| \(\Rightarrow A = -14\) | ||
| \(\Rightarrow y = x^{-2}\left\{\dfrac{2}{3}(x^3+1)^{\frac{3}{2}} - 14\right\}\) | A1 | Total: 6 |
## Question 3:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| IF is $\exp\left(\int \dfrac{2}{x}\,dx\right)$ | M1 | And with integration attempted |
| $= e^{2\ln x}$ | A1 | |
| $= x^2$ | A1 | Total: 3 | CSO **AG** be convinced |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| $\dfrac{d}{dx}\left[yx^2\right] = 3x^2(x^3+1)^{\frac{1}{2}}$ | M1A1 | PI |
| $\Rightarrow yx^2 = \dfrac{2}{3}(x^3+1)^{\frac{3}{2}} + A$ | m1 | $k(x^3+1)^{\frac{3}{2}}$ |
| | A1 | Condone missing '$A$' |
| $\Rightarrow 4 = \dfrac{2}{3}(9)^{\frac{3}{2}} + A$ | m1 | Use of boundary conditions to find constant |
| $\Rightarrow A = -14$ | | |
| $\Rightarrow y = x^{-2}\left\{\dfrac{2}{3}(x^3+1)^{\frac{3}{2}} - 14\right\}$ | A1 | Total: 6 | Any correct form |
---3
\begin{enumerate}[label=(\alph*)]
\item Show that $x ^ { 2 }$ is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
\item Solve this differential equation, given that $y = 1$ when $x = 2$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2007 Q3 [9]}}