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Question

Solve that differential equation (xy)dy=(x+y+1)dx.

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Solution

(xy)dy=(xy+1)dx
dydx=xy+1xy
Let
xy=v
Differentiating above equation w.r.t. x, we have
1dydx=dvdx
dydx=1dvdx
Therefore,
1dvdx=v+1v
dvdx=vv1v
dvdx=1v
vdv=dx
Integrating both sides, we have
vdv=dx
v22=x
v2=2x
Substituting the value of v in above equation, we have
(xy)2+2x=0
Hence the differential equation is (xy)2+2x=0.

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