| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.3 This is a straightforward separable variables question requiring standard separation, integration (including integration by parts for the x sin(x/3) term), and application of initial conditions. While it involves multiple steps and integration by parts, it follows a completely standard template with no novel insight required, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Sensibly separate variables and attempt integration of at least one side | M1 | |
| Obtain \(2y\frac{1}{3} = \ldots\) or equivalent | A1 | |
| Correct integration by parts of \(x\sin\frac{1}{3}x\) as far as \(ax\cos\frac{1}{3}x \pm \int b\cos\frac{1}{3}xdx\) | M1 | |
| Obtain \(-3x\cos\frac{1}{3}x + \int 3\cos\frac{1}{3}xdx\) or equivalent | A1 | |
| Obtain \(-3x\cos\frac{1}{3}x + 9\sin\frac{1}{3}x\) or equivalent | A1 | |
| Obtain \(y = \left(-\frac{3}{10}x\cos\frac{1}{3}x + \frac{9}{10}\sin\frac{1}{3}x + c\right)^2\) or equivalent | A1 | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(x = 0\) and \(y = 100\) to find constant | M*1 | |
| Substitute \(25\) and calculate value of \(y\) | DM*1 | |
| Obtain \(203\) | A1 | [3] |
**(i)**
Sensibly separate variables and attempt integration of at least one side | M1 |
Obtain $2y\frac{1}{3} = \ldots$ or equivalent | A1 |
Correct integration by parts of $x\sin\frac{1}{3}x$ as far as $ax\cos\frac{1}{3}x \pm \int b\cos\frac{1}{3}xdx$ | M1 |
Obtain $-3x\cos\frac{1}{3}x + \int 3\cos\frac{1}{3}xdx$ or equivalent | A1 |
Obtain $-3x\cos\frac{1}{3}x + 9\sin\frac{1}{3}x$ or equivalent | A1 |
Obtain $y = \left(-\frac{3}{10}x\cos\frac{1}{3}x + \frac{9}{10}\sin\frac{1}{3}x + c\right)^2$ or equivalent | A1 | [6]
**(ii)**
Use $x = 0$ and $y = 100$ to find constant | M*1 |
Substitute $25$ and calculate value of $y$ | DM*1 |
Obtain $203$ | A1 | [3]8 The variables $x$ and $y$ are related by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 5 } x y ^ { \frac { 1 } { 2 } } \sin \left( \frac { 1 } { 3 } x \right)$$
(i) Find the general solution, giving $y$ in terms of $x$.\\
(ii) Given that $y = 100$ when $x = 0$, find the value of $y$ when $x = 25$.
\hfill \mbox{\textit{CAIE P3 2014 Q8 [9]}}