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Maths-

General

Easy

Question

If $x fraction numerator d y over denominator d x end fraction equals y left parenthesis log space y minus log space x plus 1 right parenthesis$ then the solution of the equation is:

$\log \frac{x}{y} = c y$
$\log \frac{y}{x} = c y$
$\log \frac{x}{y} = c x$
$\log \frac{y}{x} = c x$

Hint:

In this question, we have to find the solution of the equation. For that first we will we separate both x and y and further integrate it and simplify it find the solution.

The correct answer is: $log y over x equals c x$

$x fraction numerator d y over denominator d x end fraction equals y left parenthesis log space y minus log space x plus 1 right parenthesis rightwards double arrow fraction numerator d y over denominator d x end fraction equals y over x open parentheses log y over x plus 1 close parentheses p u t space y equals v x rightwards double arrow fraction numerator d y over denominator d x end fraction equals v plus x fraction numerator d v over denominator d x end fraction rightwards double arrow v plus x fraction numerator d v over denominator d x end fraction equals v open parentheses log v plus 1 close parentheses rightwards double arrow fraction numerator d v over denominator v space log space v end fraction equals fraction numerator d x over denominator x end fraction I n t e g r a t i n g space b o t h space s i d e s comma space w e space g e t colon rightwards double arrow integral fraction numerator d v over denominator v space log space v end fraction equals integral fraction numerator d x over denominator x end fraction rightwards double arrow log left parenthesis log space v right parenthesis equals log space x plus log space c rightwards double arrow log space v equals c x rightwards double arrow log y over x equals c x$

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