| 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 (involving a square root expression) and then applying it—significantly harder than standard integrating factor problems where the form is P(x)e^∫p(x)dx. The verification requires careful differentiation of the given expression and algebraic manipulation, while part (b) involves integration with surds. However, the integrating factor is provided rather than derived from scratch, making this challenging but not at the extreme end of Further Maths difficulty. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | 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 |
## Question 7(a):
| Answer | Marks | 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 |
**Total: 4 marks**
## Question 7(b):
| Answer | Marks | 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 |
**Total: 6 marks**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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-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]}}