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Differential Equations

Question
CBSEENMA12033185

Find a one parameter family of solutions of each of the following differential equation:
                             dy over dx equals negative fraction numerator straight x plus straight y space cosx over denominator 1 plus sinx end fraction

Solution
The given differential equation is
                            dy over dx equals negative fraction numerator straight x plus straight y space cosx over denominator 1 plus sinx end fraction space space space space space or space space space dy over dx space equals negative fraction numerator straight x over denominator 1 plus sinx end fraction minus straight y fraction numerator cosx over denominator 1 plus sinx end fraction
or    dy over dx plus open parentheses fraction numerator cos space straight x over denominator 1 plus sinx end fraction close parentheses straight y space equals space minus fraction numerator straight x over denominator 1 plus sinx end fraction
Comparing it with dy over dx plus Py space equals space straight Q comma space we space get comma
                            straight P space equals space fraction numerator cosx over denominator 1 plus sinx end fraction comma space space space straight Q space equals negative fraction numerator straight x over denominator 1 plus sinx end fraction
                   integral straight P space dx space equals space integral fraction numerator cosx over denominator 1 plus sinx end fraction dx space equals space log space left parenthesis 1 plus sinx right parenthesis
therefore space space space 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 1 plus space sinx right parenthesis end exponent space equals space log space left parenthesis 1 plus sin space straight x right parenthesis
therefore  solution of differential equation is
                         straight y space straight e to the power of integral Pdx space end exponent space equals space integral straight Q space straight e to the power of integral Pdx end exponent plus straight c
or        straight y left parenthesis 1 plus sin space straight x right parenthesis space equals space integral fraction numerator negative straight x over denominator 1 plus sinx end fraction left parenthesis 1 plus sinx right parenthesis space dx plus straight c
or        straight y space straight y left parenthesis 1 plus sinx right parenthesis space equals space minus integral straight x space dx plus straight c.
or         straight y left parenthesis 1 plus sinx right parenthesis space equals space minus straight x squared over 2 plus straight c.

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