Question

If $straight x dy over dx space equals space straight y space left parenthesis log space straight y space minus space log space straight x space plus 1 right parenthesis comma$ then the solution of the equation is

$straight y space log space open parentheses straight x over straight y close parentheses space equals cx$
$space straight x space log space open parentheses straight y over straight x close parentheses space equals space cy$
$log space open parentheses straight y over straight x close parentheses space equals space cx$
$log space open parentheses x over y close parentheses space equals space cx$

Solution

log space open parentheses straight y over straight x close parentheses space equals space cx

fraction numerator straight x space dy over denominator dx end fraction space equals space straight y space left parenthesis log space straight y space minus space log space straight x plus 1 right parenthesis dy over dx space equals space straight y over straight x open parentheses log open parentheses straight y over straight x close parentheses plus 1 close parentheses put space straight y space equals vx dy over dx space equals space straight v plus space xdv over dx rightwards double arrow space straight v space plus xdv over dx space equals space straight v space left parenthesis log space straight v plus 1 right parenthesis xdv over dx space equals space straight v space log space straight v rightwards double arrow space fraction numerator dv over denominator straight v space log space straight v end fraction space equals space dx over straight x put space log space straight v equals space straight z 1 over straight v space dv space equals space dz rightwards double arrow dz over straight z space equals space dx over straight x In space straight z space equals space In space straight x space plus In space straight c straight z space equals cx log space straight v space equals space cx log space open parentheses straight y over straight x close parentheses space equals cx

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