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Question

Evaluate: dydx=y(xy)x(x+y)

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Solution

dydx=y(xy)x(x+y)
Let y=mx
dydx=m+xdmdx
Substituting in the first equation, we get
m+xdmdx=mx(xmx)x(x+mx)
m+xdmdx=x2m(1m)x2(1+m)
xdmdx=mm21+mm
xdmdx=mm2mm21+m
xdmdx=2m21+m
(1+m2m2)dm=dxx
Integrating we get,
121m2dm+121mdm=dxx
12m+12lnm=lnx+C
x2y+12lnyx=lnx+C


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