x²+y²+2xy=c
find dy/dx
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The given differential equation is…
(x2−y2)dx+2xydy=0
This can be written as…
dydx=y2−x22xy
This is the homogeneous differential equation…
To solve this type of differential equation…
we put y=vx
⇒dydx=v+xdvdx
Putting this in the given equation,then the equation reduce to…
v+xdvdx=x2(v2−1)2x2v
⇒xdvdx=v2–12v−v
⇒xdvdx=v2–1−2v22v
⇒−2vdvv2+1=dxx
⇒2vdvv2+1+dxx=0
⇒ln|v2+1|+ln|x|=ln(C)
⇒12ln|y2+x2x2|+ln|x|+ln(C)
⇒ln|x2+y2|−2ln|x|+ln|x|=ln(C)
⇒ln|x2+y2|−ln|x|=ln(C)
⇒ln|(x2+y2)x|=ln(C)
⇒x2+y2=Cx
Where 'C' being arbitrary constants.
Problem is done.
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