Let y = y(x) be the solution curve of the differential equation (x^3 - 3xy^2) d x = (y^3 - 3x^2 y) d y which passes through the point (5, 0) . If the curve intersects the line y = x 2 in the first quadrant at the point (x₀, y₀) , then the value of x₀^2 is equal to

Step-by-step solution

Given differential equation is (x^3 - 3xy^2) d x = (y^3 - 3x^2 y) d y d y d x = x^3 - 3xy^2 y^3 - 3x^2 y Let y = vx d y d x = v + x d v d x v + x d v d x = 1 - 3v^2 v^3 - 3v x d v d x = 1 - 3v^2 - v^4 + 3v^2 v^3 - 3v = 1 - v^4 v^3 - 3v v^3 - 3v 1 - v^4 d v = d x x Integrating both sides: ( v^3 1 - v^4 - 3v 1 - v^4 ) d v = d x x - 1 4 |1 - v^4| - 3 4 | 1 + v^2 1 - v^2 | = |x| + C Multiplying by -4 : |1 - v^4| + 3 | 1 + v^2 1 - v^2 | = -4 |x| + C' | (1 - v^2)(1 + v^2) (1 + v^2)^3 (1 - v^2)^3 | = (x⁻⁴) + C' | (1 + v^2