| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Verify particular integral form |
| Difficulty | Standard +0.3 This is a standard Further Maths second-order differential equation question with a given particular integral form. Students must differentiate the given form, substitute into the DE to find constants (routine algebra), solve the auxiliary equation for the complementary function, then apply boundary conditions. While it requires multiple steps and is from FP3, the method is entirely procedural with no novel insight required, making it slightly easier than average for A-level overall but typical for Further Maths content. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
3
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a , b$ and $c$ for which $a + b x + c x \mathrm { e } ^ { - 3 x }$ is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 3 x - 8 \mathrm { e } ^ { - 3 x }$$
\item Hence find the general solution of this differential equation.
\item Hence express $y$ in terms of $x$, given that $y = 1$ when $x = 0$ and that $\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow - 1$ as $x \rightarrow \infty$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q3 [12]}}