" 6."(dy)/(dx)=(1+x^(2))(1+y^(2))

If sqrt(1-x^(6))+sqrt(1-y^(6))=a(x^(3)-y^(3)), then prove that (dy)/(dx)=(x^(2))/(y^(2))sqrt((1-y^(6))/(1-x^(6)))

The solution of (dy)/(dx) = (1+y^(2))(1+x^(2))^(-1) is

y(1+x^(2))dy=x(1+y^(2))dx

y(1-x^(2))dy=x(1+y^(2))dx

The family of curves represented by (dy)/(dx)=(x^(2)+x+1)/(y^(2)+y+1) and the family represented by (dy)/(dx)+(y^(2)+y+1)/(x^(2)+x+1)=0

The family of curves represented by (dy)/(dx)=(x^(2)+x+1)/(y^(2)+y+1) and the family represented by (dy)/(dx)+(y^(2)+y+1)/(x^(2)+x+1)=0

The family of curves represented by (dy)/(dx)=(x^(2)+x+1)/(y^(2)+y+1) and the family represented by (dy)/(dx)+(y^(2)+y+1)/(x^(2)+x+1)=0

Solve the following differential equations. (i) (dy)/(dx) =(1+y^(2))/(1+x^(2)) (ii) (dy)/(dx) = (sqrt(1-y^(2)))/(sqrt(1-x^(2))) (iii) (dy)/(dx) = 2y tan hx (iv) sqrt(1+x^(2))dx + sqrt(1+y^(2))dy = 0 (v) (dy)/(dy) = e^(x-y)+x^(2)e^(-y)