Question

Show that the transformation, v =yx reduces the equation of the form yf(xy)dx + xg(xy)dy=0 to the form in which variables are separable in v and x,where f(xy)=g(xy) .Investigate the case f(xy)=g(xy)

Answer 1

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Answer 2

here if we assume y=f(xy)=g(xy), lets solve the problem backward, from
yf(xy)dx + xg(xy)dy=0 becomes
y ydx + xydy=0
if we let y=vx and dy=vdx+xdv, we get
v^2x^2 dx + vx^2(vdx+xdv) =0
v^2dx + v^2dx +xvdv=0
2v^2dx+xvdv=0
dx/x + dv/2v=0
2dx/x + dv/v=0
2lnx + lnv=ln c
lnx^2 + ln(y/x)=ln c
lnxy=ln c
xy=c