Solve the following differential equation:             f
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    Differential Equations

    Multiple Choice QuestionsShort Answer Type

    101. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
    straight e to the power of straight y dx plus straight e to the power of straight x dy space equals space 0
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    102. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
    left parenthesis sinx plus cosx right parenthesis space dy space equals space left parenthesis cosx minus sinx right parenthesis space dx
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    103. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
    straight y apostrophe space equals left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis
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    104. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
    left parenthesis straight e to the power of straight x plus straight e to the power of straight x right parenthesis space dy space minus space left parenthesis straight e to the power of straight x minus straight e to the power of straight x right parenthesis space dx space equals space 0
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    105.

    Solve the differential equation
    left parenthesis tan squared straight x plus 2 space tanx space plus 5 right parenthesis space dy over dx space equals space 2 left parenthesis 1 plus tanx right parenthesis space sec squared straight x.

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    106. Solve the following differential equation:
    x cos y dy = (x ex log x + ex) dx.
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    107. Solve the following differential equation:
    dy over dx space equals space log space left parenthesis straight x plus 1 right parenthesis
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    108. Solve the following differential equation:
                left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0
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    109. Solve the following differential equation:
    left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space straight y space dx space equals space straight x space straight y squared space dx.
                

    The given differential equation is
                         left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space straight y space dx space equals space straight x space straight y squared space dx
    or                  left parenthesis 1 minus straight x squared right parenthesis space dy space equals space left parenthesis xy squared minus straight x space straight y right parenthesis space dx space space space space space space space or space space space left parenthesis 1 minus straight x squared right parenthesis space dy space equals space straight x space left parenthesis straight y squared minus straight y right parenthesis space dx
    or                 fraction numerator 1 over denominator straight y squared minus straight y end fraction dy space equals space fraction numerator straight x over denominator 1 minus straight x squared end fraction dx
    therefore space space space space space integral fraction numerator 1 over denominator straight y squared minus straight y end fraction dy space equals space integral fraction numerator straight x over denominator 1 minus straight x squared end fraction dx therefore space space space space integral fraction numerator 1 over denominator straight y left parenthesis straight y minus 1 right parenthesis end fraction dy space equals space minus integral fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction dx therefore space space integral open square brackets fraction numerator 1 over denominator straight y left parenthesis 0 minus 1 right parenthesis end fraction plus fraction numerator 1 over denominator left parenthesis 1 right parenthesis space left parenthesis straight y minus 1 right parenthesis end fraction close square brackets space dy space equals space minus integral fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction dx therefore space integral open parentheses negative 1 over straight y plus fraction numerator 1 over denominator straight y minus 1 end fraction close parentheses space dy space equals negative integral fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction dx therefore space space space space minus log space open vertical bar straight y close vertical bar plus space log space open vertical bar straight y minus 1 close vertical bar space equals space minus space log space open vertical bar 1 minus straight x squared close vertical bar plus space log space straight c subscript 1 therefore space log space open vertical bar fraction numerator straight y minus 1 over denominator straight y end fraction close vertical bar plus space log space open vertical bar 1 minus straight x squared close vertical bar space equals space log space straight c subscript 1 therefore space log space open vertical bar open parentheses fraction numerator straight y minus 1 over denominator straight y end fraction close parentheses space left parenthesis 1 minus straight x squared right parenthesis close vertical bar space equals space log space straight c subscript 1 space space space space space space space space space therefore space space space space open vertical bar fraction numerator left parenthesis straight y minus 1 right parenthesis thin space left parenthesis 1 minus straight x squared right parenthesis over denominator straight y end fraction close vertical bar space equals space straight c subscript 1 therefore space space space space left parenthesis straight y minus 1 right parenthesis thin space left parenthesis 1 minus straight x squared right parenthesis space equals space straight c space straight y
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    110. Solve the following differential equation:
    dy over dx space equals space 1 plus straight x plus straight y plus xy comma space space straight y greater than 0

                
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