| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.2 This is a standard FP3 second-order linear differential equation with repeated roots (auxiliary equation gives m=3 twice). Part (i) is routine complementary function finding. Part (ii) tests understanding of why standard PI forms fail when they duplicate CF terms. Part (iii) requires differentiating the trial solution twice and substituting, which is algebraically involved but follows a mechanical procedure. While this is Further Maths content (inherently harder), it's a textbook example of the repeated root case with no novel problem-solving required. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
The variables $x$ and $y$ satisfy the differential equation
$$\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = e^{3x}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the complementary function. [3]
\item Explain briefly why there is no particular integral of either of the forms $y = ke^{3x}$ or $y = kxe^{3x}$. [1]
\item Given that there is a particular integral of the form $y = kx^2e^{3x}$, find the value of $k$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q5 [9]}}