Let y₁(x) and y₂(x) be two solutions of the differential equation dy dx - y x = x x for x (0, ) . It is given that y₁ ( 2 ) = a and y₂ ( 2 ) = b with a < b . If the area of the region bounded by the curves y = y₁(x) , y = y₂(x) , and the lines x = 6 and x = 3 is 3 , then the value of b - a is

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = - x and Q(x) = x x . The integrating factor (IF) is: IF = e^ - x , dx = e^ - ( x) = x The general solution is given by: y x = (x x) x , dx = x , dx = x^2 2 + C y(x) = x^2 2 x + C x For the two solutions y₁(x) and y₂(x) , their difference is: y₂(x) - y₁(x) = (C₂ - C₁) x Using the given conditions at x = 2 : y₂ ( 2 ) - y₁ ( 2 ) = b - a = (C₂ - C₁) ( 2 ) = C₂ - C₁ Thus, y₂(x) - y₁(x) = (b - a) x . The area