For the differential equation, find the general solution: from
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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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    Answer
    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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    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

    The given different equation is
                        dy over dx plus 2 straight y space equals space sin space straight x
    Comparing it with dy over dx plus straight P space straight y space space equals space straight Q comma space we space get comma space
    P = 2,   Q = sin x                       rightwards double arrow space     integral straight P space dx space equals space integral space 2 space dx space equals space 2 straight x
    straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of 2 straight x end exponent
    therefore    solution of differential equation is
               straight y. space straight e to the power of 2 straight x end exponent space equals space integral space sinx. space straight e to the power of 2 straight x end exponent space dx plus straight c space space space space space space space space space space open square brackets because space space space straight y space straight e to the power of integral Pdx 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 close square brackets

    or       straight y. space straight e to the power of 2 straight x end exponent space equals space fraction numerator 1 over denominator square root of 4 plus 1 end root end fraction straight e to the power of 2 straight x end exponent space sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 half close parentheses plus straight c

                                  open square brackets because space space integral space space straight e space to the power of ax space sin space bx space dx space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction space straight e to the power of ax. space sin space open parentheses bx minus tan to the power of negative 1 end exponent straight b over straight a close parentheses close square brackets

    or    straight y equals space fraction numerator 1 over denominator square root of 5 end fraction sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 half close parentheses plus ce to the power of negative 2 straight x end exponent
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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
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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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