Question

Find the differential equation of the curve y=2x^ab , where a and b are arbitrary constants.

Answer 1

SOLUTION

TO DETERMINE

The differential equation of the curve

$\displaystyle \sf{y = 2 {x}^{ab} }$

where a and b are arbitrary constants

EVALUATION

Here the equation of the given curve is

$\displaystyle \sf{y = 2 {x}^{ab} }$

$\implies \: \displaystyle \sf{ \frac{y}{2} = {x}^{ab} }$

Taking logarithm in both sides we get

$\displaystyle \sf{ \log \bigg( \frac{y}{2} \bigg) = \log \bigg( {x}^{ab} \bigg) }$

$\implies \displaystyle \sf{ \log y - \log 2 = ab \log x} \: \: ...(1)$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \frac{1}{y} \frac{dy}{dx} = ab . \frac{1}{x} }$

$\displaystyle \sf{ \frac{x}{y} \frac{dy}{dx} = ab }$

From Equation (1) we get

$\displaystyle \sf{ \log y - \log 2 = \log x \bigg( \frac{x}{y} \: \frac{dy}{dx} \bigg)} \:$

Which is the required differential equation

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