For the differential equation (x^2 + y^2) ,dy = xy ,dx , it is given that y(1) = 1 and y(x₀) = e , then the value of x₀ is _____

Step-by-step solution

The given differential equation is (x^2 + y^2)dy = xy dx . This can be rewritten as dx dy = x^2 + y^2 xy = x y + y x . Let x = vy , then dx dy = v + y dv dy . Substituting this into the differential equation, we get: v + y dv dy = v + 1 v y dv dy = 1 v v dv = dy y Integrating both sides, we get: v^2 2 = |y| + C Substituting v = x y : x^2 2y^2 = |y| + C Given y(1) = 1 , substituting x = 1 and y = 1 : 1 2 = (1) + C C = 1 2 The particular solution is: x^2 2y^2 = |y| + 1 2 We are given y(x₀) = e , so substituting x = x