Differential Equations

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Question

Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
$straight x space straight y apostrophe space minus space straight y space plus space straight x space sin space space open parentheses straight y over straight x close parentheses space equals 0$

Solution

The given differential equation is

xy apostrophe space minus space straight y plus space straight x space sin space open parentheses straight y over straight x close parentheses space equals space 0

therefore space space space space space space space space straight x dy over dx space equals space straight y minus straight x space sin straight y over straight x therefore space space space space space space space space dy over dx space equals space straight y over straight x minus sin straight y over straight x

Put

straight y space equals space straight v space straight x space so space that space space dy over dx space equals space straight v plus straight x dv over dx

therefore space space space space space space straight v plus straight x dv over dx space equals space straight v minus sin space straight v comma space space space space space space space space space space space space space space space space therefore space space space straight x space dv over dx space equals space minus sin space straight v

fraction numerator 1 over denominator sin space straight v end fraction dv space equals space minus 1 over straight x dx

integral space cosec space straight v space dv space equals space minus integral 1 over straight x dx

therefore space space space space space space log space space open vertical bar tan space straight v over 2 close vertical bar space equals space minus log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight c close vertical bar

log space open vertical bar straight x space tan space straight v over 2 close vertical bar space equals space log space open vertical bar straight c close vertical bar space space space space space space space space space space space space or space space space space straight x space tan straight v over 2 space equals straight c

straight x space tan space open parentheses fraction numerator straight y over denominator 2 straight x end fraction close parentheses space equals space straight c space is space the space required space solution. space