Let y=y(x) be the solution of the differential equation dy dx + y x = x , with the initial condition y ( 2 ) = 1 . Then the value of the definite integral _ /6 ^ /2 y(x) , dx is equal to

Step-by-step solution

The given differential equation is dy dx + y x = x . This is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = x and Q(x) = x . The integrating factor (I.F.) is given by: I.F. = e^ x , dx = e^ ( x) = x Multiplying the differential equation by the I.F. and integrating, we get: y x = x x , dx y x = 1 2 ^2 x + C Using the initial condition y ( 2 ) = 1 : 1 ( 2 ) = 1 2 ^2 ( 2 ) + C 1 = 1 2 + C C = 1 2 Thus, the solution curve is: y x = 1 2 ^2 x + 1 2 y(x) = 1 2 x + 1 2 x Now, we evaluate the