| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Homogeneous equation (y = vx substitution) |
| Difficulty | Challenging +1.3 This is a Further Maths FP2 question requiring substitution to transform a differential equation, then solving using separation of variables (not integrating factor despite the topic label). While it involves multiple steps and algebraic manipulation, the techniques are standard for FP2: verify a given substitution works, separate variables, integrate (likely requiring partial fractions), and find a derivative at a point. The substitution is provided rather than requiring discovery, making this a moderately challenging but routine Further Maths question—harder than typical A-level Core questions but standard for this module. |
| Spec | 4.10c Integrating factor: first order equations |
\begin{enumerate}[label=(\alph*)]
\item Show that the substitution $y = vx$ transforms the differential equation
$$3xy^2 \frac{dv}{dx} = v^4 + y^3$$ [I]
into the differential equation
$$3x v^2 \frac{dv}{dx} = 1 - 2v^3$$ [II] [3]
\item By solving differential equation (II), find a general solution of differential equation (I) in the form $y = f(x)$. [6]
\end{enumerate}
Given that $y = 2$ at $x = 1$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the value of $\frac{dy}{dx}$ at $x = 1$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q7 [11]}}