| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Solve via substitution then back-substitute |
| Difficulty | Challenging +1.3 This is a structured three-part question where part (a) requires careful but routine differentiation and substitution (5 marks of algebraic manipulation), part (b) is a standard second-order DE with constant coefficients plus particular integral (6 marks, textbook method), and part (c) is immediate from the transformation. While the initial equation looks intimidating and requires extended working, students are guided through each step with no novel insight needed—this is harder than average due to length and algebraic complexity but remains a standard Further Maths exercise. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\alph*)]
\item Show that the transformation $y = xv$ transforms the equation
$$x^2\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + (2 + 9x^2)y = x^5, \quad \text{I}$$
into the equation
$$\frac{d^2 v}{dx^2} + 9v = x^2. \quad \text{II}$$ [5]
\item Solve the differential equation II to find $v$ as a function of $x$. [6]
\item Hence state the general solution of the differential equation I. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q39 [12]}}