Find a one parameter family of solutions of each of the followin
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    Differential Equations

    Multiple Choice QuestionsLong Answer Type

    261.

    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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    262.

    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

      

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    Answer
    263.

    Solve:  open parentheses fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction close parentheses dx over dy equals 1 comma space space space straight x not equal to space 0.

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    264. Solve the following differential equation:
    dy over dx space plus 2 space straight y space tan space straight x space equals space sin space straight x
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    265. For the differential equation, find the general solution:
    dy over dx plus 2 straight y space equals sin space straight x
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    266. Solve the following differential equation:
    sin space straight x space dy over dx plus straight y space cosx space equals space cos space straight x space sin squared straight x
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    Multiple Choice QuestionsShort Answer Type

    267. Find a one parameter family of solutions of each of the following differential equation:
    straight y apostrophe cos squared straight x space equals space tanx minus space straight y
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    268. 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

    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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    Multiple Choice QuestionsLong Answer Type

    269.

    Solve:
    straight y apostrophe minus 2 straight y space equals space cos space 3 straight x

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    Answer

    Multiple Choice QuestionsShort Answer Type

    270.

    Solve:
    straight y apostrophe plus straight y space equals space sin space straight x

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    Answer
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