Question

The solution curves to the DE 2xyy¹= y²-x² are a) the circles passing through the origin with centers on x axis. b) the ellipse. c) the hyperbolas. d) the parabolas​

Answer 1

dy=2xyy2−x2

Substitute y=vx

∴v+xdxdv=2vv2−1⇒xdxdv=−2v1+v2⇒1+v22vdv+xdx=0

Integrating log(1+v2)+logx=logk

∴x(1+v2)=k.

Substitute v=xy

⇒x²+y²=kx.