| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| 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 homogeneous differential equation requiring the y=vx substitution (shown in part a), separation of variables, and partial fractions. While it involves multiple techniques and is from FP2, the method is algorithmic and well-practiced. The multi-part structure guides students through the solution, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.10c Integrating factor: first order equations |
\begin{enumerate}
\item (a) Show that the substitution $y = v x$ transforms the differential equation
\end{enumerate}
$$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 3 } + y ^ { 3 }$$
into the differential equation
$$3 v ^ { 2 } x \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1 - 2 v ^ { 3 }$$
(b) By solving differential equation (II), find a general solution of differential equation (I) in the form $y = \mathrm { f } ( x )$.
Given that $y = 2$ at $x = 1$,\\
(c) find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $x = 1$\\
\hfill \mbox{\textit{Edexcel FP2 2012 Q7 [11]}}