| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Substitution reducing to first order linear ODE |
| Difficulty | Standard +0.8 This is a Further Pure 3 differential equations question requiring a non-obvious substitution (given) followed by solving a separable equation and back-substitution. While the substitution is provided, students must carefully apply the chain rule and algebraic manipulation (4 marks), then solve the resulting separable DE and express the final answer correctly (5 marks). The multi-step nature, algebraic complexity, and FP3 context place this moderately above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)4.10c Integrating factor: first order equations |
The variables $x$ and $y$ are related by the differential equation
$$x^3 \frac{dy}{dx} = xy + x + 1. \qquad (A)$$
\begin{enumerate}[label=(\roman*)]
\item Use the substitution $y = u - \frac{1}{x}$, where $u$ is a function of $x$, to show that the differential equation may be written as
$$x^2 \frac{du}{dx} = u.$$ [4]
\item Hence find the general solution of the differential equation (A), giving your answer in the form $y = f(x)$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q5 [9]}}