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Differential Equations

Question
CBSEENMA12033129
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Find a one parameter family of solutions of each of the following differential equation:
(y2 – 2xy) dx = (x2 – 2xy) dy

Solution
The given differential equation is
                       (y2 – 2xy) dx = (x2 – 2xy) dy
or               dy over dx space equals fraction numerator straight y squared minus 2 xy over denominator straight x squared minus 2 xy end fraction
Put y = v x so that dy over dx space equals space straight v plus straight x dv over dx
therefore space space space space space space space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared straight x squared minus 2 vx squared over denominator straight x squared minus 2 vx squared end fraction therefore space space space space space space space straight v plus straight x dv over dx space equals fraction numerator straight v squared minus 2 straight v over denominator 1 minus 2 straight v end fraction therefore space space space space space space space space space space straight x dv over dx space equals space fraction numerator straight v squared minus 2 straight v over denominator 1 minus 2 straight v end fraction minus straight v space equals space fraction numerator straight v squared minus 2 straight v minus straight v plus 2 straight v squared over denominator 1 minus 2 straight v end fraction therefore space space space space space space space straight x dv over dx space equals space fraction numerator 3 straight v squared minus 3 straight v over denominator 1 minus 2 straight v end fraction
Separating the variables and integrating, we get,
                   integral fraction numerator 1 minus 2 straight v over denominator 3 straight v squared minus 3 straight v end fraction equals space space integral 1 over straight x dx space space or space space 1 third integral fraction numerator 1 minus 2 straight v over denominator straight v squared minus straight v end fraction dv space equals integral 1 over straight x dx
therefore space space space space minus 1 third space log space left parenthesis straight v squared minus straight v right parenthesis space equals space log space straight x plus space log space straight c therefore space space space space log space left parenthesis straight v squared minus straight v right parenthesis to the power of fraction numerator negative 1 over denominator 3 end fraction end exponent space equals space log space cx space therefore space space space space space space space space space space space left parenthesis straight v squared minus straight v right parenthesis to the power of negative 1 third end exponent space equals space straight c space straight x therefore space space space space space space space open parentheses straight y squared over straight x squared minus straight y over straight x close parentheses to the power of negative 1 third end exponent space equals space space straight c space straight x therefore space space space space space space open parentheses straight y squared minus straight x space straight y close parentheses to the power of negative 1 third end exponent space equals space cx space cross times space straight x to the power of fraction numerator negative 2 over denominator 3 end fraction end exponent therefore space space space space space space space space left parenthesis straight y squared minus straight x space straight y right parenthesis to the power of fraction numerator negative 1 over denominator 3 end fraction end exponent space equals space straight c space straight x to the power of 1 third end exponent therefore space space space space space space space space space left parenthesis straight y squared minus straight x space straight y right parenthesis to the power of negative 1 end exponent space equals space Ax therefore space space space space space space fraction numerator 1 over denominator straight y squared minus xy end fraction space equals space Ax or space space space xy squared minus straight x squared straight y space equals space straight c comma space which space is space required space solution. space      

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