| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Standard +0.8 This FP3 question requires recognizing the integrating factor form, applying a non-trivial partial fraction decomposition or logarithm result from the formula book, then performing algebraic manipulation with fractional powers and integration. While systematic, it demands careful handling of multiple techniques and algebraic fluency beyond standard A-level, typical of Further Maths material but not requiring deep conceptual insight. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules4.10c Integrating factor: first order equations |
The differential equation
$$\frac{dy}{dx} + \frac{1}{1 - x^2} y = (1 - x)^{\frac{1}{2}}, \quad \text{where } |x| < 1,$$
can be solved by the integrating factor method.
\begin{enumerate}[label=(\roman*)]
\item Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as $\left(\frac{1 + x}{1 - x}\right)^{\frac{1}{2}}$. [2]
\item Hence find the solution of the differential equation for which $y = 2$ when $x = 0$, giving your answer in the form $y = f(x)$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q4 [8]}}