| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Standard +0.3 This is a structured Further Maths question with clear guidance (integrating factor given, initial condition provided). Part (a) is routine chain rule practice. Part (b)(i) requires verification rather than discovery of the integrating factor. Part (b)(ii) involves standard integration techniques. While the nested logarithm adds mild complexity, the step-by-step scaffolding and standard method make this slightly easier than average for FP3 material. |
| Spec | 1.07l Derivative of ln(x): and related functions4.10c Integrating factor: first order equations |
5
\begin{enumerate}[label=(\alph*)]
\item Differentiate $\ln ( \ln x )$ with respect to $x$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\ln x$ is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { x \ln x } y = 9 x ^ { 2 } , \quad x > 1$$
\item Hence find the solution of this differential equation, given that $y = 4 \mathrm { e } ^ { 3 }$ when $x = \mathrm { e }$.\\
(6 marks)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q5 [9]}}