| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 This is a standard Further Maths second-order linear differential equation with constant coefficients and a constant non-homogeneous term. The auxiliary equation yields complex roots (routine calculation), finding the particular integral is straightforward (try y=constant), and applying initial conditions involves basic algebra. While it's a Further Maths topic, it follows a completely algorithmic procedure with no conceptual challenges, making it slightly easier than average overall. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((m+2)^2 = -1\) | M1 | |
| \(m = -2 \pm i\) | A1 | |
| CF is \(e^{-2x}(A\cos x + B\sin x)\) or \(e^{-A+ix} + Be^{-1-ix}\) | M1, A1/ | |
| PI try \(y = p \Rightarrow 5p = 5\) PI is \(y = 1\) | B1 | |
| GS \(y = e^{-2x}(\text{Acos}x + B\sin x) + 1\) | B1/ | Total: 6 |
| (b) \(x=0, y = 2 \Rightarrow A = 1\) | B1/ | |
| \(y'(x) = -2e^{-2x}(\text{A}\cos x+B\sin x) + e^{-2x}(-A\sin x+B\cos x)\) | M1, A1/ | |
| \(y'(0) = 3 \Rightarrow 3 = -2A+B \Rightarrow B = 5\) | A1/ | Total: 4 |
| \(y = e^{-2x}(\cos x + 5\sin x) + 1\) |
**(a)** $(m+2)^2 = -1$ | M1 | | Completing sq or formula
$m = -2 \pm i$ | A1 | |
CF is $e^{-2x}(A\cos x + B\sin x)$ or $e^{-A+ix} + Be^{-1-ix}$ | M1, A1/ | | If m is real give M0; Ft on wrong $a$'s and $b$'s but roots must be complex
PI try $y = p \Rightarrow 5p = 5$ PI is $y = 1$ | B1 | |
GS $y = e^{-2x}(\text{Acos}x + B\sin x) + 1$ | B1/ | Total: 6 | Their CF + their PI with two arbitrary constants
**(b)** $x=0, y = 2 \Rightarrow A = 1$ | B1/ | | Provided previous B1/ awarded
$y'(x) = -2e^{-2x}(\text{A}\cos x+B\sin x) + e^{-2x}(-A\sin x+B\cos x)$ | M1, A1/ | | Product rule used
$y'(0) = 3 \Rightarrow 3 = -2A+B \Rightarrow B = 5$ | A1/ | Total: 4 | Ft on one slip
$y = e^{-2x}(\cos x + 5\sin x) + 1$ | | |3
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5$$
\item Hence express $y$ in terms of $x$, given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3$ when $x = 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q3 [10]}}