| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Topic | Differential equations |
| Type | Separable variables - partial fractions |
| Difficulty | Challenging +1.2 This is a separable differential equation requiring partial fractions (decomposing 1/(y(1+y))), integration, and applying initial conditions to find the constant. Part (ii) requires analyzing the derivative's sign, which is straightforward given the factored form. While it involves multiple techniques and careful algebra, it follows a standard Further Maths procedure without requiring novel insight or particularly complex manipulation. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.08k Separable differential equations: dy/dx = f(x)g(y) |
\begin{enumerate}[label=(\roman*)]
\item Solve the differential equation
$$\frac{dy}{dx} = y(1 + y)(1 - x),$$
given that $y = 1$ when $x = 1$. Give your answer in the form $y = f(x)$, where $f$ is a function to be determined. [7]
\item By considering the sign of $\frac{dy}{dx}$ near $(1, 1)$, or otherwise, show that this point is a maximum point on the curve $y = f(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q13 [10]}}