Differential Equations

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Question

Solve $open parentheses straight x space sin space straight y over straight x close parentheses dy space equals open parentheses straight y space sin space straight y over straight x minus straight x close parentheses dx$

Solution

The given differential equation is

open parentheses straight x space sin space straight y over straight x close parentheses dy space equals space open parentheses ysin straight y over straight x minus straight x close parentheses dx space space space space space space or space space space space dy over dx equals fraction numerator ysin begin display style straight y over straight x end style minus straight x over denominator straight x space sin begin display style straight y over straight x end style end fraction space space space space space... left parenthesis 1 right parenthesis

Put y = v x so that

dy over dx equals straight v plus straight x dv over dx

therefore space space left parenthesis 1 right parenthesis space becomes comma space space straight v plus straight x dv over dx equals fraction numerator straight v space straight x space sin space straight v space minus space straight x over denominator straight x space sin space straight v end fraction

straight v plus straight x dv over dx equals fraction numerator straight v space sinv minus 1 over denominator sin space straight v end fraction space space space or space space space straight x dv over dx equals fraction numerator straight v space sinv minus 1 over denominator sin space straight v end fraction minus straight v space space space or space space straight x dv over dx equals negative fraction numerator 1 over denominator sin space straight v end fraction

Separating the variables,

sin space straight v space dv space equals space minus 1 over straight x dx

Integrating,

negative cos space straight v space equals space minus log space open vertical bar straight x close vertical bar space minus space straight c space space space or space cos space open parentheses straight y over straight x close parentheses space equals space log space open vertical bar straight x close vertical bar space plus straight c

which is required solution.