Pre-U Pre-U 9794/2 2012 June — Question 8 6 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2012
SessionJune
Marks6
TopicDifferential equations
TypeSeparable variables - standard (polynomial/exponential x-side)
DifficultyModerate -0.3 This is a straightforward separable differential equation requiring only variable separation, integration of power functions, and application of an initial condition. The technique is standard and the integration is routine (∫y^(-2)dy and ∫x^3dx), making it slightly easier than average despite being worth 6 marks.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)
Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]
AnswerMarks Guidance
Separate variables, prior to integration: \(\int \frac{r-1}{y^2}dy = \int x^3 dx\)M1, A1
\(\frac{1}{y} = \frac{1}{4}x^4 \quad (+c)\)A1, A1, M1
Subs into expression including \(c\) and solve: \(c = \frac{4}{4}\) so \(y = \frac{4}{x^4+1}\) AEFA1 [6] [6] 
Separate variables, prior to integration: $\int \frac{r-1}{y^2}dy = \int x^3 dx$ | M1, A1 |

$\frac{1}{y} = \frac{1}{4}x^4 \quad (+c)$ | A1, A1, M1 |

Subs into expression including $c$ and solve: $c = \frac{4}{4}$ so $y = \frac{4}{x^4+1}$ AEF | A1 [6] | [6]
Solve the differential equation $\frac{dy}{dx} = -y^2 x^3$, where $y = 2$ when $x = 1$, expressing your solution in the form $y = f(x)$. [6]

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q8 [6]}}
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