Question

x²+y²+2xy=c
find dy/dx

Answer 1

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.