| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | With preliminary integration |
| Difficulty | Standard +0.8 Part (a) requires recognizing how to apply the substitution effectively with integration by parts, which is non-trivial. Part (b) is a standard integrating factor problem but requires dividing by x first to get standard form, then recognizing the integrating factor is x³, and finally using the result from part (a). The connection between parts and the multi-step nature elevates this above a routine FP2 question, though it remains within standard Further Maths territory. |
| Spec | 1.08h Integration by substitution4.10c Integrating factor: first order equations |
\begin{enumerate}[label=(\alph*)]
\item Using the substitution $t = x^2$, or otherwise, find
$$\int x^3 e^{-x^2} \, dx.$$ [6]
\item Find the general solution of the differential equation
$$x\frac{dy}{dx} + 3y = xe^{-x^2}, \quad x > 0.$$ [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q19 [10]}}