Question

Find one-parameter families of solution curves of the following differential equation:
$straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0 space space space left parenthesis straight x not equal to 0 right parenthesis$

Solution

The given differential equation is
   $straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0$
or $straight x dy over dx plus left parenthesis 1 plus space straight x space cot space straight x right parenthesis space straight y space equals straight x$
or    $dy over dx plus open parentheses 1 over straight x plus cot space straight x close parentheses. straight y space equals space 1$
Comparing it with $dy over dx plus Py space equals straight Q comma space space we space get comma space space space straight P space equals space 1 over straight x plus cot space straight x comma space space straight Q space equals space 1$
$integral straight P space dx space equals space integral open parentheses 1 over straight x plus cot space straight x close parentheses space dx space equals space log space straight x plus space log space sinx space equals space log space left parenthesis straight x space sin space straight x right parenthesis$
$therefore space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space left parenthesis straight x space sinx right parenthesis end exponent space equals space space straight x space sin space straight x$
Solution of given differential equation is
$straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c$
or $straight y. space left parenthesis straight x space sin space straight x right parenthesis space equals space integral space 1. space straight x space sinx space dx space plus straight c$
or    $straight x space straight y space sin space straight x space equals integral space straight x space sin space straight x space dx space plus space straight c$
or $straight x space straight y space sin space straight x space equals straight x. space left parenthesis negative cos space straight x right parenthesis minus space integral 1. space left parenthesis negative cos space straight x right parenthesis space dx plus space straight c$
or    $straight x space straight y space sin space straight x space equals space minus straight x space cos space straight x space space plus space integral space cosx space dx plus straight c$
or    $straight x space straight y space sin space straight x space equals space minus straight x space cos space straight x space plus space sin space straight x space plus straight c$
or    $straight y equals negative cot space straight x plus 1 over straight x plus fraction numerator straight c over denominator straight x space sin space straight x end fraction$
which is required solution.

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