Let y = y(x) be the solution of the differential equation dy dx + 2xy = y^2 e^ x^2 x , satisfying y(0) = 1 2 . Then the value of the integral _ - /3 ^ /3 e^ -x^2 y(x) dx is equal to:

Step-by-step solution

The given differential equation is: dy dx + 2xy = y^2 e^ x^2 x Dividing the entire equation by y^2 , we get a Bernoulli differential equation: 1 y^2 dy dx + 2x y = e^ x^2 x Let u = 1 y , which implies du dx = - 1 y^2 dy dx . Substituting this into the equation gives: - du dx + 2x u = e^ x^2 x du dx - 2x u = -e^ x^2 x This is a linear differential equation with integrating factor (I.F.) = e^ -2x dx = e^ -x^2 . Multiplying by the I.F., the equation becomes: d dx (u e^ -x^2 ) = - x Integrating both sides with respect