y(z_x)+x(z_y)+z=y

Use Method of characteristics

a=y

b=x

d-cz=y-z

Thus

dx/y=dy/b=dz/(y-z)

Taking first and second term

xdx=ydy

x^2-y^2=A

x=sqrt(y^2+A)

My question is, at this stage of the calculation, must we account for a negative constant A such that

x=sqrt(y^2-A)

and if not why not?

Thank you