| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2004 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Challenging +1.2 This is a standard second-order linear ODE with constant coefficients requiring complementary function (complex roots), particular integral (trial solution for sin 2x), and asymptotic analysis. Part (b) requires recognizing that the exponentially decaying CF terms vanish for large x, leaving only the PI. While multi-step, all techniques are routine FP2 material with no novel insight required. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
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$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x , x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation.
\item Show that for large values of $x$ this general solution may be approximated by a sine function and find this sine function.\\
(3)(Total 12 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2004 Q4 [12]}}