y^2=(1-x)/(1+x); differentiating:
2yy'=(-(1+x)-(1-x))/(1+x)^2=(-1-x-1+x)/(1+x)^2=-2/(1+x)^2.
yy'+1/(1+x)^2=0; yy'(1+x)^2+1=0.
But y=√(1-x^2)/(1+x) when we multiply ((1-x)/(1+x))^(1/2) by ((1+x)/(1+x))^(1/2).
So we have yy'(1+x)^2+1=y'(1+x)√(1-x^2)+1=0. But 1+x=√(1-x^2)/y, so y'(1-x^2)/y+1=0 and (1-x^2)dy/dx+y=0 QED.