| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Combined polynomial and exponential RHS |
| Difficulty | Standard +0.3 This is a standard Further Maths second-order differential equation question with routine steps: verifying a given particular integral by substitution, finding the complementary function from characteristic equation (roots 1 and 4), and applying initial conditions. All techniques are textbook procedures with no novel insight required, making it slightly easier than average for FP3. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
1 It is given that $y$ satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 8 x - 10 - 10 \cos 2 x$$
\begin{enumerate}[label=(\alph*)]
\item Show that $y = 2 x + \sin 2 x$ is a particular integral of the given differential equation.
\item Find the general solution of the differential equation.
\item Hence express $y$ in terms of $x$, given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q1 [11]}}