| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable with partial fractions |
| Difficulty | Standard +0.3 This is a straightforward separable differential equation requiring partial fractions (a standard technique), separation of variables, integration, and applying initial conditions. Part (iii) requires basic limit analysis. All steps are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to express integrand in partial fractions, e.g. obtain \(A\) or \(B\) in \(\frac{A}{y} + \frac{B}{4-y}\) | M1 | |
| Obtain \(\frac{1}{4}\left(\frac{1}{y} + \frac{1}{4-y}\right)\), or equivalent | A1 | |
| Integrate and obtain \(\frac{1}{4}\ln y - \frac{1}{4}\ln(4-y)\), or equivalent | A1√ + A1√ | 4 |
| (ii) Separate variables correctly, integrate \(\frac{A}{y} + \frac{B}{4-y}\) and obtain further term \(x\), or equivalent | M1* | |
| Use \(y = 1\) and \(x = 0\) to evaluate a constant, or as limits | M1(dep*) | |
| Obtain answer in any correct form | A1 | |
| Obtain final answer \(y = 4/(3e^{-4x} + 1)\), or equivalent | A1 | 4 |
| (iii) State that \(y\) approaches \(4\) as \(x\) becomes very large | B1 | 1 |
**(i)** Attempt to express integrand in partial fractions, e.g. obtain $A$ or $B$ in $\frac{A}{y} + \frac{B}{4-y}$ | M1 |
Obtain $\frac{1}{4}\left(\frac{1}{y} + \frac{1}{4-y}\right)$, or equivalent | A1 |
Integrate and obtain $\frac{1}{4}\ln y - \frac{1}{4}\ln(4-y)$, or equivalent | A1√ + A1√ | 4
**(ii)** Separate variables correctly, integrate $\frac{A}{y} + \frac{B}{4-y}$ and obtain further term $x$, or equivalent | M1* |
Use $y = 1$ and $x = 0$ to evaluate a constant, or as limits | M1(dep*) |
Obtain answer in any correct form | A1 |
Obtain final answer $y = 4/(3e^{-4x} + 1)$, or equivalent | A1 | 4
**(iii)** State that $y$ approaches $4$ as $x$ becomes very large | B1 | 18 (i) Using partial fractions, find
$$\int \frac { 1 } { y ( 4 - y ) } \mathrm { d } y$$
(ii) Given that $y = 1$ when $x = 0$, solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 4 - y ) ,$$
obtaining an expression for $y$ in terms of $x$.\\
(iii) State what happens to the value of $y$ if $x$ becomes very large and positive.
\hfill \mbox{\textit{CAIE P3 2005 Q8 [9]}}