Question

Find one-parameter families of solution curves of the following differential equation:
$left parenthesis 1 plus straight x squared right parenthesis space dy space plus space 2 xy space dx space equals space cot space straight x space dx space space space space left parenthesis straight x not equal to 0 right parenthesis$

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Solution

The given differential equation is

left parenthesis 1 plus straight x squared right parenthesis space dy plus space 2 xy space dx space equals space cot space straight x space dx

open parentheses 1 plus straight x squared close parentheses dy over dx plus 2 space straight x space straight y space equals space space cot space straight x

dy over dx plus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction. space straight y space equals space fraction numerator cot space straight x over denominator 1 plus straight x squared end fraction

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get comma space straight P equals space fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction comma space space straight Q space equals space fraction numerator cot space straight x over denominator 1 plus straight x squared end fraction

integral straight P space dx space equals space integral fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction dx space equals space log space left parenthesis 1 plus straight x squared right parenthesis. 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 1 plus straight x right parenthesis end exponent space equals space 1 plus straight x squared

Solution of given differential equation is

ye 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

straight y left parenthesis 1 plus straight x squared right parenthesis space equals space integral open parentheses fraction numerator cot space straight x over denominator 1 plus straight x squared end fraction close parentheses. space left parenthesis 1 plus straight x squared right parenthesis space dx plus straight c

or space space space straight y left parenthesis 1 plus straight x squared right parenthesis space equals space integral cot space straight x space dx plus straight c or space space straight y left parenthesis 1 plus straight x squared right parenthesis space equals space log space open vertical bar sin space straight x close vertical bar space plus straight c space or space space space space space space space space space space straight y space equals space left parenthesis 1 plus straight x squared right parenthesis to the power of negative 1 end exponent space log space open vertical bar sin space straight x close vertical bar space plus space straight c space left parenthesis 1 plus straight x squared right parenthesis to the power of negative 1 end exponent which space is space required space solution. space

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