| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Homogeneous equation (y = vx substitution) |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 differential equations question requiring a given substitution followed by separation of variables. Part (a) is routine verification (3 marks), part (b) requires separating variables and integrating (standard technique, 7 marks), and part (c) applies initial conditions (2 marks). While it's a multi-step problem requiring careful algebra and the substitution back to find y=f(x), it follows a well-established template for homogeneous differential equations that FP2 students practice extensively. The substitution is given rather than requiring insight to find it, making this moderately above average difficulty but not requiring novel problem-solving. |
| Spec | 4.10c Integrating factor: first order equations |
\begin{enumerate}[label=(\alph*)]
\item Show that the substitution $y = vx$ transforms the differential equation
$$\frac{dy}{dx} = \frac{x}{y} + \frac{3y}{x}, x > 0, y > 0$$ (I)
into the differential equation $x\frac{dv}{dx} = 2v + \frac{1}{v}$. (II) (3)
\item By solving differential equation (II), find a general solution of differential equation (I) in the form $y = f(x)$. (7)
\end{enumerate}
Given that $y = 3$ at $x = 1$, (c)find the particular solution of differential equation (I).(2)
\hfill \mbox{\textit{Edexcel FP2 2008 Q7}}