| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring recognition that tan x is the integrating factor (non-trivial verification involving derivative of tan x and coefficient matching), followed by standard integration techniques including integration by parts for tan²x. The non-standard form and the need to manipulate trigonometric expressions elevates this above typical C1-C4 differential equations, but it follows a clear method once the integrating factor is verified. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^24.10c Integrating factor: first order equations |
5
\begin{enumerate}[label=(\alph*)]
\item Show that $\tan x$ is an integrating factor for the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { \sec ^ { 2 } x } { \tan x } y = \tan x$$
(2 marks)
\item Hence solve this differential equation, given that $y = 3$ when $x = \frac { \pi } { 4 }$.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q5 [8]}}