| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Substitution z = x + y or similar linear combination |
| Difficulty | Standard +0.8 This is a Further Pure 3 differential equation requiring a non-standard substitution technique. While the substitution is given, students must correctly apply the chain rule (dz/dx = 1 + dy/dx), perform algebraic manipulation, then separate variables and integrate. The integration involves partial fractions or logarithms. It's more challenging than standard A-level separable equations but follows a guided structure, placing it moderately above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)4.10a General/particular solutions: of differential equations |
\begin{enumerate}[label=(\roman*)]
\item Use the substitution $z = x + y$ to show that the differential equation
$$\frac{dy}{dx} = \frac{x + y + 3}{x + y - 1} \qquad (A)$$
may be written in the form $\frac{dz}{dx} = \frac{2(z + 1)}{z - 1}$. [3]
\item Hence find the general solution of the differential equation (A). [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q3 [7]}}