Let x = x(y) be the solution of the differential equation (1 - y^2) dx - (xy + 3y^2 1 - y^2 ) dy = 0 , with the initial condition x(0) = 0 . Then the value of x ( 1 2 ) is equal to

Step-by-step solution

Given differential equation is (1 - y^2) dx - (xy + 3y^2 1 - y^2 ) dy = 0 . Rearranging the terms, we get: dx dy - y 1 - y^2 x = 3y^2 1 - y^2 This is a linear differential equation of the form dx dy + P(y)x = Q(y) , where P(y) = - y 1 - y^2 . Integrating Factor (I.F.) = e^ - y 1 - y^2 dy Let 1 - y^2 = t -2y dy = dt -y dy = dt 2 I.F. = e^ 1 2t dt = e^ 1 2 (1 - y^2) = 1 - y^2 Multiplying the differential equation by the I.F., the solution is given by: x I.F. = Q(y) I.F. dy + C x 1 - y^2 = 3y^2 1 - y^2 1 - y^2 dy + C