| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2024 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Integrating factor with non-standard form |
| Difficulty | Challenging +1.8 This Further Maths question requires verifying a non-standard integrating factor (not the usual exponential form) and applying it. The verification involves differentiating a complex expression and algebraic manipulation to show it works, followed by integration requiring substitution. While technically demanding with multiple steps, the integrating factor is given, making it more procedural than requiring deep insight. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dy}{dx} + \frac{y}{\sqrt{x^2+16}} = x\) | B1 | Divides through by \(\sqrt{x^2+16}\) |
| \(e^{\int \frac{1}{\sqrt{x^2+16}}dx} = e^{\sinh^{-1}(\frac{x}{4})}\) | M1A1 | Finds integrating factor |
| \(= \frac{1}{4}x + \frac{1}{4}\sqrt{x^2+16}\) | A1 | AG |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{d}{dx}\left(y\left(x+\sqrt{x^2+16}\right)\right) = x^2 + x\sqrt{x^2+16}\) | M1A1 | Correct form on LHS and RHS |
| \(y\left(x+\sqrt{x^2+16}\right) = \frac{1}{3}x^3 + \frac{1}{3}\left(x^2+16\right)^{\frac{3}{2}} + C\) | M1A1 | Integrates RHS. RHS of the correct form |
| \(6\left(3+\sqrt{25}\right) = \frac{27}{3} + \frac{25}{3}\sqrt{25} + C\) | M1 | Substitutes initial conditions into their expression |
| \(y\left(x+\sqrt{x^2+16}\right) = \frac{1}{3}x^3 + \frac{1}{3}\left(x^2+16\right)^{\frac{3}{2}} - \frac{8}{3}\) | A1 | OE |
| 6 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{dx} + \frac{y}{\sqrt{x^2+16}} = x$ | B1 | Divides through by $\sqrt{x^2+16}$ |
| $e^{\int \frac{1}{\sqrt{x^2+16}}dx} = e^{\sinh^{-1}(\frac{x}{4})}$ | M1A1 | Finds integrating factor |
| $= \frac{1}{4}x + \frac{1}{4}\sqrt{x^2+16}$ | A1 | AG |
| | **4** | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{d}{dx}\left(y\left(x+\sqrt{x^2+16}\right)\right) = x^2 + x\sqrt{x^2+16}$ | M1A1 | Correct form on LHS and RHS |
| $y\left(x+\sqrt{x^2+16}\right) = \frac{1}{3}x^3 + \frac{1}{3}\left(x^2+16\right)^{\frac{3}{2}} + C$ | M1A1 | Integrates RHS. RHS of the correct form |
| $6\left(3+\sqrt{25}\right) = \frac{27}{3} + \frac{25}{3}\sqrt{25} + C$ | M1 | Substitutes initial conditions into their expression |
| $y\left(x+\sqrt{x^2+16}\right) = \frac{1}{3}x^3 + \frac{1}{3}\left(x^2+16\right)^{\frac{3}{2}} - \frac{8}{3}$ | A1 | OE |
| | **6** | |7
\begin{enumerate}[label=(\alph*)]
\item Show that an appropriate integrating factor for
$$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$
is $\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }$ .\\
\includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
\item Hence find the solution of the differential equation
$$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$
for which $y = 6$ when $x = 3$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q7 [10]}}