| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.8 This is a separable differential equation requiring multiple steps: separation of variables, integration involving both polynomial and logarithmic terms, applying two boundary conditions to find both the constant k and the integration constant, then rearranging to express x in terms of t. The algebra is non-trivial and the two-condition setup adds complexity beyond standard single-condition problems, but the techniques are all standard A-level material. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables correctly and integrate at least one side | M1 | |
| Obtain term in \(l\), or equivalent | B1 | |
| Obtain term of the form \(a \ln(k - x^3)\) | M1 | |
| Obtain term \(-\frac{2}{3}\ln(k - x^3)\), or equivalent | A1 | |
| EITHER: Evaluate a constant or use limits \(t = 1, x = 1\) in a solution containing \(c \ln l\) and \(b \ln(k - x^3)\) | M1* | |
| Obtain correct answer in any form e.g. \(\ln t = -\frac{2}{3}\ln(k - x^3) + \frac{2}{3}\ln(k - 1)\) | A1 | |
| Use limits \(t = 4, x = 2\), and solve for \(k\) | M1 (dep*) | |
| Obtain \(k = 9\) | A1 | |
| OR: Using limits \(t = 1, x = 1\) and \(t = 4, x = 2\) in a solution containing \(c \ln l\) and \(b \ln(k - x^3)\) obtain an equation in \(k\) | M1* | |
| Obtain a correct equation in any form, e.g. \(\ln 4 = -\frac{2}{3}\ln(k - 8) + \frac{2}{3}\ln(k - 1)\) | A1 | |
| Solve for \(k\) | M1 (dep*) | |
| Obtain \(k = 9\) | A1 | |
| Substitute \(k = 9\) and obtain \(x = (9 - 8t^{-\frac{3}{1}})^{\frac{1}{3}}\) | A1 | [9 marks] |
| (ii) State that \(x\) approaches \(9^{\frac{1}{3}}\), or equivalent | B1 | [1 mark] |
**(i)** Separate variables correctly and integrate at least one side | M1 |
Obtain term in $l$, or equivalent | B1 |
Obtain term of the form $a \ln(k - x^3)$ | M1 |
Obtain term $-\frac{2}{3}\ln(k - x^3)$, or equivalent | A1 |
EITHER: Evaluate a constant or use limits $t = 1, x = 1$ in a solution containing $c \ln l$ and $b \ln(k - x^3)$ | M1* |
Obtain correct answer in any form e.g. $\ln t = -\frac{2}{3}\ln(k - x^3) + \frac{2}{3}\ln(k - 1)$ | A1 |
Use limits $t = 4, x = 2$, and solve for $k$ | M1 (dep*) |
Obtain $k = 9$ | A1 |
OR: Using limits $t = 1, x = 1$ and $t = 4, x = 2$ in a solution containing $c \ln l$ and $b \ln(k - x^3)$ obtain an equation in $k$ | M1* |
Obtain a correct equation in any form, e.g. $\ln 4 = -\frac{2}{3}\ln(k - 8) + \frac{2}{3}\ln(k - 1)$ | A1 |
Solve for $k$ | M1 (dep*) |
Obtain $k = 9$ | A1 |
Substitute $k = 9$ and obtain $x = (9 - 8t^{-\frac{3}{1}})^{\frac{1}{3}}$ | A1 | [9 marks]
**(ii)** State that $x$ approaches $9^{\frac{1}{3}}$, or equivalent | B1 | [1 mark]
---8 The variables $x$ and $t$ satisfy the differential equation
$$t \frac { \mathrm {~d} x } { \mathrm {~d} t } = \frac { k - x ^ { 3 } } { 2 x ^ { 2 } }$$
for $t > 0$, where $k$ is a constant. When $t = 1 , x = 1$ and when $t = 4 , x = 2$.\\
(i) Solve the differential equation, finding the value of $k$ and obtaining an expression for $x$ in terms of $t$.\\
(ii) State what happens to the value of $x$ as $t$ becomes large.
\hfill \mbox{\textit{CAIE P3 2013 Q8 [10]}}