| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Challenging +1.2 Part (a) is a standard second-order linear ODE with constant coefficients requiring complementary function (repeated root λ=-5) and particular integral using undetermined coefficients for sin t. Part (b) requires recognizing that the complementary function e^(-5t) terms decay to zero for large t, leaving only the particular integral, then expressing it in R sin(t-φ) form. While this involves multiple techniques and some conceptual understanding of asymptotic behavior, it follows a well-established procedure taught in Further Maths courses with no novel insight required. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(m^2 + 10m + 25 = 0 \Rightarrow m = -5\) | M1 | Auxiliary equation |
| \(x = e^{-5t}(At + B)\) | A1 | Complementary function; allow '\(x=\)' missing |
| \(x = p\sin t + q\cos t \Rightarrow x' = p\cos t - q\sin t \Rightarrow x'' = -p\sin t - q\cos t\) | M1 A1 | Particular integral and its derivatives; both derivatives must be present and of correct form (only allow sign errors) |
| \(-p - 10q + 25p = 338 \quad -q + 10p + 25q = 0\) | M1 | Substitutes and equates coefficients |
| \(p = 12 \quad q = -5\) | A1 | |
| \(x = e^{-5t}(At + B) + 12\sin t - 5\cos t\) | A1 | Must have '\(x=\)' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x \approx 12\sin t - 5\cos t = 13\left(\frac{12}{13}\sin t - \frac{5}{13}\cos t\right)\) | M1 | Considering PI only |
| \(12\sin t - 5\cos t = 13\sin(t - 0.395\ldots)\) | A1 A1 | A1 for \(R\) and A1 for \(\phi\); condone \(22.6°\); withhold final A1 if working with incorrect PI; CWO |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2 + 10m + 25 = 0 \Rightarrow m = -5$ | M1 | Auxiliary equation |
| $x = e^{-5t}(At + B)$ | A1 | Complementary function; allow '$x=$' missing |
| $x = p\sin t + q\cos t \Rightarrow x' = p\cos t - q\sin t \Rightarrow x'' = -p\sin t - q\cos t$ | M1 A1 | Particular integral and its derivatives; both derivatives must be present and of correct form (only allow sign errors) |
| $-p - 10q + 25p = 338 \quad -q + 10p + 25q = 0$ | M1 | Substitutes and equates coefficients |
| $p = 12 \quad q = -5$ | A1 | |
| $x = e^{-5t}(At + B) + 12\sin t - 5\cos t$ | A1 | Must have '$x=$' |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x \approx 12\sin t - 5\cos t = 13\left(\frac{12}{13}\sin t - \frac{5}{13}\cos t\right)$ | M1 | Considering PI only |
| $12\sin t - 5\cos t = 13\sin(t - 0.395\ldots)$ | A1 A1 | A1 for $R$ and A1 for $\phi$; condone $22.6°$; withhold final A1 if working with incorrect PI; CWO |
---5
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 338 \sin t$$
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-10_2715_35_143_2012}
\item Show that, for large positive values of $t$ and for any initial conditions,
$$x \approx R \sin ( t - \phi ) ,$$
where the constants $R$ and $\phi$ are to be determined.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q5 [10]}}