| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| 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 recognizing the non-standard form, verifying a given integrating factor involving √(x²-1), then applying it to solve the DE with an initial condition. The algebraic manipulation with nested radicals and the verification process demand careful technique beyond standard integrating factor problems, but the integrating factor is provided in part (a), reducing the problem-solving burden. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} + \frac{1}{\sqrt{x^2-1}}\,y = \frac{x^2 - x\sqrt{x^2-1}}{\sqrt{x^2-1}}\) | B1 | Divides through by \(\sqrt{x^2-1}\) |
| \(e^{\int \frac{1}{\sqrt{x^2-1}}\,dx} = e^{\cosh^{-1}x}\) | M1 A1 | Finds integrating factor. M1 for correct form \(e^{a\int \frac{1}{\sqrt{x^2-1}}\,dx}\) where \(a\) is a non-zero constant |
| \(x + \sqrt{x^2-1}\) | A1 | AG |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d}{dx}\!\left(y\!\left(x + \sqrt{x^2-1}\right)\right) = \frac{x}{\sqrt{x^2-1}}\) | M1 A1 | Correct form on LHS and simplifies RHS |
| \(y\!\left(x + \sqrt{x^2-1}\right) = \sqrt{x^2-1} + C\) | M1 A1 | Integrates RHS. For M1, needs to integrate non-zero multiple of \(\frac{x}{\sqrt{x^2-1}}\) |
| \(2 = \frac{3}{4} + C\) leading to \(C = \frac{5}{4}\) | M1 | Substitutes initial conditions |
| \(y = \dfrac{\sqrt{x^2-1} + \frac{5}{4}}{x + \sqrt{x^2-1}}\) | M1 A1 | Divides through by coefficient of \(y\) |
| 7 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} + \frac{1}{\sqrt{x^2-1}}\,y = \frac{x^2 - x\sqrt{x^2-1}}{\sqrt{x^2-1}}$ | B1 | Divides through by $\sqrt{x^2-1}$ |
| $e^{\int \frac{1}{\sqrt{x^2-1}}\,dx} = e^{\cosh^{-1}x}$ | M1 A1 | Finds integrating factor. M1 for correct form $e^{a\int \frac{1}{\sqrt{x^2-1}}\,dx}$ where $a$ is a non-zero constant |
| $x + \sqrt{x^2-1}$ | A1 | AG |
| | **4** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d}{dx}\!\left(y\!\left(x + \sqrt{x^2-1}\right)\right) = \frac{x}{\sqrt{x^2-1}}$ | M1 A1 | Correct form on LHS and simplifies RHS |
| $y\!\left(x + \sqrt{x^2-1}\right) = \sqrt{x^2-1} + C$ | M1 A1 | Integrates RHS. For M1, needs to integrate non-zero multiple of $\frac{x}{\sqrt{x^2-1}}$ |
| $2 = \frac{3}{4} + C$ leading to $C = \frac{5}{4}$ | M1 | Substitutes initial conditions |
| $y = \dfrac{\sqrt{x^2-1} + \frac{5}{4}}{x + \sqrt{x^2-1}}$ | M1 A1 | Divides through by coefficient of $y$ |
| | **7** | |7
\begin{enumerate}[label=(\alph*)]
\item Show that an appropriate integrating factor for
$$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$
is $x + \sqrt { x ^ { 2 } - 1 }$.
\item Hence find the solution of the differential equation
$$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$
for which $y = 1$ when $x = \frac { 5 } { 4 }$. Give your answer in the form $y = f ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q7 [11]}}