Find the general solution of each of the following equation: fr
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    Differential Equations

    Multiple Choice QuestionsShort Answer Type

    111. Solve the following differential equation:
    open parentheses straight x squared straight y plus straight x squared close parentheses space dx space plus space left parenthesis straight x space straight y squared minus straight y squared right parenthesis space dy space equals space 0

                
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    112. Solve the following differential equation:
    straight y left parenthesis 1 minus straight x squared right parenthesis dy over dx space equals space straight x left parenthesis 1 plus straight y squared right parenthesis


                
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    113. Find the general solution of each of the following equations
    dy over dx equals straight x to the power of 5 plus straight x squared minus 2 over straight x
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    114. Find the general solution of each of the following equations:
    left parenthesis straight x plus 2 right parenthesis space dy over dx space equals space straight x squared plus 4 straight x minus 9
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    115. Find the general solution of each of the following equations:
    dy over dx plus 2 straight x equals space straight e to the power of 3 straight x end exponent
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    116. Find the general solution of each of the following equations:
    dy over dx equals fraction numerator straight y left parenthesis straight x squared minus 1 right parenthesis over denominator straight x left parenthesis straight y squared minus 1 right parenthesis end fraction
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    117. Find the general solution of each of the following equation:
    left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space 2 space straight x cubed space straight y space where space straight y apostrophe space equals space dy over dx

    The given differential equation is
                    left parenthesis straight x minus 1 right parenthesis space straight y apostrophe space equals space space 2 straight x cubed straight y
    or     open parentheses straight x minus 1 close parentheses space dy over dx space equals space 2 straight x cubed straight y space space space space space space space space space space space space space or space space space space space dy over dx space equals space fraction numerator 2 straight x cubed straight y over denominator straight x minus 1 end fraction.
    Separating the variables, we get,
                        right enclose table row cell 1 over straight y dy space equals space fraction numerator 2 straight x cubed over denominator straight x minus 1 end fraction dx end cell row cell or space space 1 over straight y dy space equals space open parentheses 2 straight x squared plus 2 straight x plus 2 plus fraction numerator 2 over denominator straight x minus 1 end fraction close parentheses dx end cell row cell table row cell Integrating comma space we space get comma end cell row cell log space open vertical bar straight y close vertical bar space equals space 2 straight x cubed over 3 plus fraction numerator 2 straight x squared over denominator 2 end fraction plus 2 straight x end cell row cell table row cell plus 2 space log space open vertical bar straight x minus 1 close vertical bar plus straight A end cell row cell or space log space open vertical bar straight y close vertical bar minus 2 space log space open vertical bar straight x minus 1 close vertical bar end cell row cell equals 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A end cell end table end cell end table end cell end table end enclose space table row cell table row cell table row cell table row cell stack straight x minus 1 long division enclose space 2 straight x cubed end enclose with 2 straight x squared plus 2 straight x plus 2 on top end cell row cell space space space space space space 2 straight x cubed minus 2 straight x squared end cell row cell negative space space plus space end cell end table end cell row _________ row cell 2 space straight x squared end cell end table end cell row cell 2 straight x squared minus 2 straight x end cell row cell negative space space space plus end cell end table end cell row ___________ row cell table row cell 2 space straight x end cell row cell 2 straight x minus 2 end cell row cell table row cell negative space plus end cell row __________ row 2 end table end cell end table end cell end table space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
    or  log space open vertical bar fraction numerator straight y over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction close vertical bar space equals 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A space space space space space space rightwards double arrow space space space open vertical bar fraction numerator straight y over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction close vertical bar space equals space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x plus straight A end exponent
    or                  straight y space equals straight e to the power of straight x space left parenthesis straight x minus 1 right parenthesis squared space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x end exponent space space space space or space space space space straight y space equals straight e space left parenthesis straight x minus 1 right parenthesis squared space straight e to the power of 2 over 3 straight x cubed plus straight x squared plus 2 straight x end exponent
    where e is an arbitrary constant. 
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    118. Find the general solution of the differential equation:
    dy over dx plus square root of fraction numerator 1 minus straight y squared over denominator 1 minus straight x squared end fraction end root space equals space 0.
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    Answer
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    119. Find a one parameter family of solutions of each of the following differential equation:
    y' = ex + y + e–x + y
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    120. Find a one parameter family of solutions of each of the following differential equation:
    y' = (cos2x – sin2x) cos2 y
    97 Views
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