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Differential Equations

Question
CBSEENMA12033128
Wired Faculty App

Find a one parameter family of solutions of each of the following differential equation:
y2 + x2 y' = x y y'

Solution
The given differential equation is
                   straight y squared plus straight x squared dy over dx equals space straight x. space straight y space dy over dx
or         left parenthesis straight x space straight y space minus space straight x squared right parenthesis space dy over dx space equals straight y squared
therefore space space space space space space space space space dy over dx space equals space fraction numerator straight y squared over denominator xy minus straight y squared end fraction
Put straight y space equals space straight v space straight x space so space that space dy over dx space equals straight v plus straight x dv over dx
therefore space space space space straight v plus straight x dv over dx equals space fraction numerator straight v squared minus straight x squared over denominator straight x squared straight v minus straight x squared end fraction therefore space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared over denominator straight v minus 1 end fraction therefore space space space space space space space space straight x dv over dx space equals fraction numerator straight v squared over denominator straight v minus 1 end fraction minus straight v therefore space space space space space space straight x dv over dx space equals space fraction numerator straight v over denominator straight v minus 1 end fraction therefore space space space space space fraction numerator straight v minus 1 over denominator straight v end fraction dv space equals space 1 over straight x dx therefore space space space space integral open parentheses 1 minus 1 over straight v close parentheses space dv space equals space integral 1 over straight x dx therefore space space space space space straight v minus log space straight v space equals space logx plus log space straight c therefore space space space space straight y over straight x minus log space straight y over straight x space equals space log space straight x space plus space log space straight c therefore space space space space straight y over straight x minus log space straight y space equals space log space straight c 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 space space space space space space space space space space space space rightwards double arrow space space straight y over straight x space equals space log space straight c space straight y therefore space space space space space space space space space straight c space straight y space equals space straight e to the power of straight y over straight x end exponent space is space the space required space solution. space

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