Find the general solution of xy*(1+y^2)dx - (1+x^2)dy = 0?

Answer 1

collect like terms on one side
xdx/(1+x^2)=dy/[(1+y^2)*y]

1/2* 2x/(1+x^2)= dy/y-1/2*2y/(1+y^2)

By integrating each side,
1/2*ln(1+x^2)
=ln|y|-1/2*ln(1+y^2)

use log rules, take square of both sides,

C*y^2 / (1+y^2)=1+x^2

y=sqrt([1+x^2]/[C-1-x^2])

Answer 2

Write the eqn. :
xdx/(1+x^2)=dy/y*(1+y^2)
2xdx/(1+x^2)=2ydy/y^2*(1+y^2)
now put [1+x^2]=z & y^2=t. Then a very simple integration follows. Hence finish the problem.