| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Topic | First order differential equations (integrating factor) |
| Type | Separable with partial fractions |
| Difficulty | Standard +0.3 This is a standard separable differential equation with guided steps through partial fractions. Part (i) is routine A-level algebra, part (ii) follows directly from separation and integration, and part (iii) involves straightforward substitution and rearrangement. The question structure provides significant scaffolding, making it slightly easier than average despite involving multiple techniques. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(2\ln y - 2\ln | 1-y | = x + C\) aef — A1 |
**(i)** Attempt use of the form $\frac{A}{y} + \frac{B}{1-y}$ and remove fractions — M1
Obtain $A = 2$ — A1
Obtain $B = 2$ — A1 **[3]**
**(ii)** Attempt to separate variables and use result from (i) — M1
Attempt integration of both sides — M1
Obtain $2\ln y - 2\ln|1-y| = x + C$ aef — A1
Attempt use of at least one log law correctly — M1
State or imply $\ln\left(\frac{y^2}{(1-y)^2}\right) = x + C$ and obtain convincingly
$\frac{y^2}{(1-y)^2} = Ae^x$ AG — A1 **[5]**
**(iii)** Substitute $(0, 2)$ and obtain $A = 4$ — B1
Select the correct root of $-2$ and attempt to make $y$ the subject — M1
i.e. $\frac{y}{(1-y)} = -2e^{\frac{x}{2}}$
Obtain $y = \frac{2e^{\frac{x}{2}}}{2e^{\frac{x}{2}}-1}$ or equivalent simplified form — A1 **[3]**11 A differential equation is given by $2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )$.\\
(i) Express $\frac { 2 } { y ( 1 - y ) }$ in partial fractions.\\
(ii) Hence show by integration that $\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }$.\\
(iii) Given that $x = 0$ when $y = 2$, find the value of $A$ and express $y$ in terms of $x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2014 Q11 [11]}}