Differential Equations

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Question

Find one-parameter families of solution curves of the following differential equation:
$dy over dx minus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction comma space space straight y space equals space straight x squared plus 2$

Solution

The given differential equation is

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

Comparing it with

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

therefore space space space space space integral straight P space dx space equals space minus integral fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction dx equals negative log space left parenthesis 1 plus straight x squared right parenthesis therefore space space space space space straight e to the power of integral space Pdx end exponent space equals space straight e to the power of negative log space left parenthesis 1 plus straight x squared right parenthesis end exponent space equals space straight e to the power of log space left parenthesis 1 plus straight x squared right parenthesis to the power of negative 1 end exponent end exponent space equals space fraction numerator 1 over denominator 1 plus straight x squared end fraction

therefore

solution of differential equation is

straight y space straight e to the power of integral straight P space dx end exponent space equals integral straight Q. space straight e to the power of integral straight P space dx end exponent dx space plus space straight c space space space or space space straight y. space fraction numerator 1 over denominator 1 plus straight x squared end fraction space equals space integral left parenthesis straight x squared plus 2 right parenthesis. space fraction numerator 1 over denominator straight x squared plus 1 end fraction dx plus straight c

fraction numerator straight y over denominator 1 plus straight x squared end fraction space equals integral fraction numerator straight x squared plus 2 over denominator straight x squared plus 1 end fraction dx plus straight c

fraction numerator straight y over denominator 1 plus straight x squared end fraction space equals space integral open parentheses 1 plus fraction numerator 1 over denominator straight x squared plus 1 end fraction close parentheses dx plus straight c

fraction numerator straight y over denominator 1 plus straight x squared end fraction space equals space straight x plus tan to the power of negative 1 end exponent straight x plus straight c