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The given differential equation is <img class=Wirisformula style=vertical-align: -25px; src=/application/zrc/images/qvar/MAEN12063937-1.png alt=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 width=357 height=55 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mfrac»«mi mathvariant=¨normal¨»y«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«/mrow»«/mfenced»«mi»dy«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfenced»«mrow»«mi»ysin«/mi»«mfrac»«mi mathvariant=¨normal¨»y«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«mo»-«/mo»«mi mathvariant=¨normal¨»x«/mi»«/mrow»«/mfenced»«mi»dx«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mfrac»«mi»dy«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»ysin«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»y«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«/mstyle»«mo»-«/mo»«mi mathvariant=¨normal¨»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»y«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«/mstyle»«/mrow»«/mfrac»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»1«/mn»«mo»)«/mo»«/math»> Put y = v x so that <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12063937-2.png alt=dy over dx equals straight v plus straight x dv over dx width=86 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»dy«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»x«/mi»«mfrac»«mi»dv«/mi»«mi»dx«/mi»«/mfrac»«/math»> <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12063937-3.png alt=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 width=242 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8756;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mn»1«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mi»becomes«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»x«/mi»«mfrac»«mi»dv«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«/mrow»«/mfrac»«/math»> or <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12063937-4.png alt=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 width=393 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»v«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»x«/mi»«mfrac»«mi»dv«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mi»sinv«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«/mrow»«/mfrac»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mfrac»«mi»dv«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mi»sinv«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«/mrow»«/mfrac»«mo»-«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mfrac»«mi»dv«/mi»«mi»dx«/mi»«/mfrac»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«/mrow»«/mfrac»«/math»> Separating the variables, <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12063937-5.png alt=sin space straight v space dv space equals space minus 1 over straight x dx width=106 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»sin«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mi»dv«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«mi»dx«/mi»«/math»> Integrating, <img class=Wirisformula style=vertical-align: -10px; src=/application/zrc/images/qvar/MAEN12063937-6.png alt=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 width=276 height=28 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mi»log«/mi»«mo»§#160;«/mo»«mfenced open=¨|¨ close=¨|¨»«mi mathvariant=¨normal¨»x«/mi»«/mfenced»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»c«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mfenced»«mfrac»«mi mathvariant=¨normal¨»y«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«/mfenced»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»log«/mi»«mo»§#160;«/mo»«mfenced open=¨|¨ close=¨|¨»«mi mathvariant=¨normal¨»x«/mi»«/mfenced»«mo»§#160;«/mo»«mo»+«/mo»«mi mathvariant=¨normal¨»c«/mi»«/math»> which is required solution.

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

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.

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