Edexcel A-Level Mathematics A2 P4.6.4 Analytical Solution of Simple First Questions

Practise solving separable first order differential equations and using initial conditions to form models in x, y, h or V.

Exam points

separate variables and integrate both sides to form a general solution
use initial values to find constants in models for depth, current, height or volume
solve the model for a time or value, giving exact or 3 significant figure answers

[Maximum number: 4]

Solve the differential equation (II) to determine v as a function of x

$\frac{1}{v^{2}} \frac{\mathrm{~d} v}{\mathrm{~d} x}=-2 x \Rightarrow \int v^{-2} \mathrm{~d} v=-2 \int x \mathrm{~d} x$ $\Rightarrow \frac{v^{-1}}{-1}=-2 \frac{x^{2}}{2}(+c)$ $\Rightarrow \frac{1}{v}=x^{2}+c$ $\Rightarrow v=\frac{1}{x^{2}+c}$
(4)

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Edexcel A-Level Mathematics A2 P4.6.4 Analytical Solution of Simple First Questions question 1