The general solution of the differential equation d y d x = x y x^2+y^2 is

Step-by-step solution

The given differential equation is dy dx = xy x^2 + y^2 . Taking the reciprocal, we have dx dy = x^2 + y^2 xy = x y + y x . Let x = vy , then dx dy = v + y dv dy . Substituting into the equation: v + y dv dy = v + 1 v . y dv dy = 1 v . v dv = dy y . Integrating both sides: v dv = dy y v^2 2 = |y| + C . Substituting v = x y : x^2 2y^2 = |y| + C . Rearranging for |y| : |y| = x^2 2y^2 - C . Taking the exponential of both sides: y = e^ x^2 2y^2 - C = e^ -C e^ x^2 2y^2 . Letting e^ -C = c , we obtain y = c e^ x^2 2y^2 .