Solve the given differential equation by using an appropriate substitution.
41. y2dx+(x2+xy+y2)dy=0, y(0)=1
x=vy
dx=vdy+ydv
y2[vdy+ydv]+[v2y2+vy2+y2]dy=0
vy2dy+y3dv+v2y2dy+vy2dy+y2dy=0
2vy2dy+v2y2dy+y2dy+y3dv=0
y2[2v+v2+1]dy+y2(y)dv=0
y2[2v+v2+1]dy= -y2(y)dv
, where (x/y)=v
Multiply through by (x+y).
, when x=0, y=1
-----(Am I doing this right, and do I have to go further???)
The answer in my textbook says
(x+y)ln(y)+x=0