| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - variable coefficients |
| Difficulty | Standard +0.3 This is a standard integrating factor question from Further Maths with straightforward application of the method. Part (a) requires identifying the integrating factor (2x+1)^4, multiplying through, and integrating a polynomial. Part (b) simply applies the boundary condition dy/dx=0 at x=0 to find the constant. While it's Further Maths content, the execution is mechanical with no conceptual challenges or novel insights required. |
| Spec | 4.10c Integrating factor: first order equations |
4
\begin{enumerate}[label=(\alph*)]
\item By using an integrating factor, find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 } { 2 x + 1 } y = 4 ( 2 x + 1 ) ^ { 5 }$$
giving your answer in the form $y = \mathrm { f } ( x )$.
\item The gradient of a curve at any point $( x , y )$ on the curve is given by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 ( 2 x + 1 ) ^ { 5 } - \frac { 4 } { 2 x + 1 } y$$
The point whose $x$-coordinate is zero is a stationary point of the curve. Using your answer to part (a), find the equation of the curve.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q4 [10]}}