| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Verify particular integral form |
| Difficulty | Standard +0.3 This is a standard second-order differential equation problem from Further Maths. Part (a) involves straightforward substitution and algebraic manipulation to find k. Part (b) requires finding the complementary function (standard auxiliary equation with complex roots) and applying initial conditions. While it's Further Maths content, the techniques are routine and well-practiced, making it slightly easier than average for A-level overall but typical for FP3. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-k\sin x + 2k\cos x + 5k\sin x = 8\sin x + 4\cos x\) | M1, A1 | Differentiation and substitution into DE |
| \(k = 2\) | A1 | Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Auxiliary equation: \(m^2 + 2m + 5 = 0\) | ||
| \(m = \frac{-2 \pm \sqrt{4-20}}{2}\) | M1 | Formula or completing the square, PI |
| \(m = -1 \pm 2i\) | A1 | |
| CF: \(\{y_C\} = e^{-x}(A\sin 2x + B\cos 2x)\) | A1F | ft provided \(m\) is not real |
| GS: \(\{y\} = e^{-x}(A\sin 2x + B\cos 2x) + k\sin x\) | B1F | ft on CF + PI; must have 2 arbitrary constants |
| When \(x=0\), \(y=1 \Rightarrow B=1\) | B1F | |
| \(\frac{dy}{dx} = -e^{-x}(A\sin 2x + B\cos 2x) + e^{-x}(2A\cos 2x - 2B\sin 2x) + k\cos x\) | M1 | Product rule |
| When \(x=0\), \(\frac{dy}{dx} = 4 \Rightarrow 4 = -B + 2A + k\) | A1 | PI |
| \(\Rightarrow A = \frac{3}{2}\) | ||
| \(y = e^{-x}\left(\frac{3}{2}\sin 2x + \cos 2x\right) + 2\sin x\) | A1 | Total: 8 marks. CSO |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-k\sin x + 2k\cos x + 5k\sin x = 8\sin x + 4\cos x$ | M1, A1 | Differentiation and substitution into DE |
| $k = 2$ | A1 | Total: 3 marks |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Auxiliary equation: $m^2 + 2m + 5 = 0$ | | |
| $m = \frac{-2 \pm \sqrt{4-20}}{2}$ | M1 | Formula or completing the square, PI |
| $m = -1 \pm 2i$ | A1 | |
| CF: $\{y_C\} = e^{-x}(A\sin 2x + B\cos 2x)$ | A1F | ft provided $m$ is not real |
| GS: $\{y\} = e^{-x}(A\sin 2x + B\cos 2x) + k\sin x$ | B1F | ft on CF + PI; must have 2 arbitrary constants |
| When $x=0$, $y=1 \Rightarrow B=1$ | B1F | |
| $\frac{dy}{dx} = -e^{-x}(A\sin 2x + B\cos 2x) + e^{-x}(2A\cos 2x - 2B\sin 2x) + k\cos x$ | M1 | Product rule |
| When $x=0$, $\frac{dy}{dx} = 4 \Rightarrow 4 = -B + 2A + k$ | A1 | PI |
| $\Rightarrow A = \frac{3}{2}$ | | |
| $y = e^{-x}\left(\frac{3}{2}\sin 2x + \cos 2x\right) + 2\sin x$ | A1 | Total: 8 marks. CSO |5 It is given that $y$ satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 8 \sin x + 4 \cos x$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of the constant $k$ for which $y = k \sin x$ is a particular integral of the given differential equation.
\item Solve the differential equation, expressing $y$ in terms of $x$, given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4$ when $x = 0$.\\
(8 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2009 Q5 [11]}}