| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2023 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Modeling context with interpretation |
| Difficulty | Challenging +1.3 This is a coupled differential equations problem from Further Maths requiring elimination to form a second-order ODE, solving it using auxiliary equations, then finding the corresponding solution for y. While it involves multiple steps and Further Maths content (making it harder than standard A-level), the techniques are fairly standard: differentiation, substitution, solving characteristic equations, and applying initial conditions. The elimination step in (a)(i) is routine, and the auxiliary equation factors nicely to give integer roots (1 and 8), making the solution straightforward. The main challenge is managing the algebra across multiple parts rather than requiring deep insight. |
| Spec | 4.10h Coupled systems: simultaneous first order DEs |
Two species of insects, $X$ and $Y$, co-exist on an island. The populations of the species at time $t$ years are $x$ and $y$ respectively, where $x$ and $y$ are measured in millions. The situation can be modelled by the differential equations
$$\frac{\mathrm{d}x}{\mathrm{d}t} = 3x + 10y,$$
$$\frac{\mathrm{d}y}{\mathrm{d}t} = x + 6y.$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} - 9\frac{\mathrm{d}x}{\mathrm{d}t} + 8x = 0$.
\item Find the general solution for $x$ in terms of $t$. [7]
\end{enumerate}
\item Find the corresponding general solution for $y$. [4]
\item When $t = 0$, $\frac{\mathrm{d}x}{\mathrm{d}t} = 5$ and the population of species $Y$ is 4 times the population of species $X$. Find the particular solution for $x$ in terms of $t$. [6]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2023 Q13 [17]}}