| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Combined polynomial and exponential RHS |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring method of undetermined coefficients for a particular integral with mixed RHS (polynomial + exponential), finding complementary function, and applying boundary conditions including a limit condition. While systematic, it demands multiple techniques and careful algebraic manipulation beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
2
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $p$ and $q$ for which $p + q x \mathrm { e } ^ { - 2 x }$ is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 4 - 9 \mathrm { e } ^ { - 2 x }$$
\item Hence find the general solution of this differential equation.
\item Hence express $y$ in terms of $x$, given that $y = 4$ when $x = 0$ and that $\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow 0$ as $x \rightarrow \infty$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2011 Q2 [12]}}