| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | First order differential equations (integrating factor) |
| Type | Bernoulli equation |
| Difficulty | Challenging +1.2 This is a structured Bernoulli equation problem where part (i) guides students through the substitution, removing the main conceptual hurdle. Part (ii) requires applying an integrating factor and back-substitution—standard Further Maths techniques. The scaffolding in part (i) significantly reduces difficulty compared to an unguided Bernoulli equation, placing this moderately above average difficulty. |
| Spec | 4.10c Integrating factor: first order equations |
**(i)** $u = \frac{1}{y^3} \Rightarrow \frac{\mathrm{d}u}{\mathrm{d}x} = -\frac{3}{y^4}\times\frac{\mathrm{d}y}{\mathrm{d}x}$ **B1**
Then $\frac{\mathrm{d}y}{\mathrm{d}x} + y = 3xy^4$ becomes $-\frac{3}{y^4}\cdot\frac{\mathrm{d}y}{\mathrm{d}x} - \frac{3}{y^3} = -9x \Rightarrow \frac{\mathrm{d}u}{\mathrm{d}x} - 3u = -9x$ **AG** **M1 A1**
**[3]**
**(ii) Method 1**
IF is $\mathrm{e}^{-3x}$ **M1 A1**
$\Rightarrow u\mathrm{e}^{-3x} = \int -9x\mathrm{e}^{-3x}\,\mathrm{d}x$ **M1**
$= 3x\mathrm{e}^{-3x} - \int 3\mathrm{e}^{-3x}\,\mathrm{d}x$ Use of "parts" **M1**
$= (3x+1)\mathrm{e}^{-3x} + C$ **A1**
Gen. Soln. is $u = 3x + 1 + C\mathrm{e}^{3x}$ **ft** **B1**
$\Rightarrow y^3 = \frac{1}{3x+1+C\mathrm{e}^{3x}}$ **ft** **B1**
Using $x = 0$, $y = \frac{1}{2}$ to find $C$: $C = 7$ or $y^3 = \frac{1}{3x+1+7\mathrm{e}^{3x}}$ **M1 A1**
**[9]**9 (i) Show that the substitution $u = \frac { 1 } { y ^ { 3 } }$ transforms the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }$ into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x$$
(ii) Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }$, given that $y = \frac { 1 } { 2 }$ when $x = 0$. Give your answer in the form $y ^ { 3 } = \mathrm { f } ( x )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2012 Q9 [9]}}