| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | With preliminary integration |
| Difficulty | Challenging +1.8 This is a Further Maths first-order DE requiring integrating factor method (standard technique) combined with a non-trivial integration given in part (a). The integrating factor is straightforward (1/x), but students must recognize how to use the preliminary result involving hyperbolic functions and apply the boundary condition. The combination of hyperbolic functions, product rule verification, and multi-step solution with substitution elevates this above routine A-level questions but remains within expected Further Maths territory. |
| Spec | 4.07f Inverse hyperbolic: logarithmic forms4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d}{dx}\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3}\right)\) | M1 | Applies product rule and \(\frac{d}{dx}\left(\cosh^{-1}\frac{x}{3}\right) = \frac{1}{\sqrt{x^2-9}}\) |
| \(\frac{x^2}{2\sqrt{x^2-9}} + \frac{1}{2}\sqrt{x^2-9} - \frac{9}{2\sqrt{x^2-9}}\) | A1 | OE |
| \(\frac{x^2-9}{2\sqrt{x^2-9}} + \frac{1}{2}\sqrt{x^2-9} = \sqrt{x^2-9}\) | A1 | Puts over common denominator, AG |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} - \frac{y}{x} = x\sqrt{x^2-9}\) | B1 | Divides through by \(x\) |
| \(e^{-\int x^{-1}d\theta} = e^{-\ln x} = x^{-1}\) | M1 A1 | Finds integrating factor |
| \(\frac{d}{dx}(x^{-1}y) = \sqrt{x^2-9}\) | M1 | Correct form on LHS, \(\frac{d}{dx}(Iy)\) for their integrating factor \(I\) |
| \(x^{-1}y = \frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3} + C\) | M1 A1 | Integrates RHS. For M1, RHS must be of the form \(c\sqrt{x^2-9}\), where \(c\) is a non-zero constant |
| \(\frac{1}{3} = C\) | M1 | Substitutes initial conditions |
| \(y = x\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3} + \frac{1}{3}\right)\) | M1 A1 | Divides through by their integrating factor. Accept \(y = x\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\ln\left(\frac{x}{3} + \frac{1}{3}\sqrt{x^2-9}\right) + \frac{1}{3}\right)\) |
| 9 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d}{dx}\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3}\right)$ | **M1** | Applies product rule and $\frac{d}{dx}\left(\cosh^{-1}\frac{x}{3}\right) = \frac{1}{\sqrt{x^2-9}}$ |
| $\frac{x^2}{2\sqrt{x^2-9}} + \frac{1}{2}\sqrt{x^2-9} - \frac{9}{2\sqrt{x^2-9}}$ | **A1** | OE |
| $\frac{x^2-9}{2\sqrt{x^2-9}} + \frac{1}{2}\sqrt{x^2-9} = \sqrt{x^2-9}$ | **A1** | Puts over common denominator, AG |
| | **3** | |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} - \frac{y}{x} = x\sqrt{x^2-9}$ | **B1** | Divides through by $x$ |
| $e^{-\int x^{-1}d\theta} = e^{-\ln x} = x^{-1}$ | **M1 A1** | Finds integrating factor |
| $\frac{d}{dx}(x^{-1}y) = \sqrt{x^2-9}$ | **M1** | Correct form on LHS, $\frac{d}{dx}(Iy)$ for their integrating factor $I$ |
| $x^{-1}y = \frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3} + C$ | **M1 A1** | Integrates RHS. For M1, RHS must be of the form $c\sqrt{x^2-9}$, where $c$ is a non-zero constant |
| $\frac{1}{3} = C$ | **M1** | Substitutes initial conditions |
| $y = x\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3} + \frac{1}{3}\right)$ | **M1 A1** | Divides through by their integrating factor. Accept $y = x\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\ln\left(\frac{x}{3} + \frac{1}{3}\sqrt{x^2-9}\right) + \frac{1}{3}\right)$ |
| | **9** | |
---7
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \frac { x } { 2 } \sqrt { x ^ { 2 } - 9 } - \frac { 9 } { 2 } \cosh ^ { - 1 } \frac { x } { 3 } \right) = \sqrt { x ^ { 2 } - 9 }$$
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_67_1579_413_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_77_1581_497_322}
\item Find the solution of the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = x ^ { 2 } \sqrt { x ^ { 2 } - 9 }$$
given that $y = 1$ when $x = 3$. Give your answer in the form $y = \mathrm { f } ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_2716_35_143_2012}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q7 [12]}}