| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Homogeneous equation (y = vx substitution) |
| Difficulty | Challenging +1.2 This is a structured multi-part differential equations question requiring the substitution y=vx, separation of variables, and back-substitution. While it involves several steps and algebraic manipulation, the question provides explicit guidance at each stage (gives the substitution, tells you what to obtain, and asks you to verify the final form). The techniques are standard for FP2 level, though the algebraic manipulation in part (a) and integration in part (b) require care. It's moderately harder than average A-level due to being Further Maths content with multi-step reasoning, but the scaffolding prevents it from being truly challenging. |
| Spec | 4.10c Integrating factor: first order equations |
\begin{enumerate}[label=(\alph*)]
\item Use the substitution $y = vx$ to transform the equation
$$\frac{dy}{dx} = \frac{(4x + y)(x + y)}{x^2}, \quad x > 0 \quad \text{(I)}$$
into the equation
$$x\frac{dv}{dx} = (2 + v)^2. \quad \text{(II)}$$ [4]
\item Solve the differential equation II to find $v$ as a function of $x$. [5]
\item Hence show that
$$y = -2x - \frac{x}{\ln x + c}, \text{ where } c \text{ is an arbitrary constant,}$$
is a general solution of the differential equation I. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q14 [10]}}