Let y=y(x) be the solution of the differential equation dy dx - y = e^x x , with y(0) = 0 . The area bounded by the curve y=y(x) , the x -axis, and the ordinates x=0 and x= 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) = -1 and Q(x) = e^x x . Integrating Factor (I.F.) = e^ -1 dx = e^ -x Multiplying the equation by the I.F. and integrating, we get: y e^ -x = (e^x x) e^ -x dx y e^ -x = x dx y e^ -x = - x + C Using the initial condition y(0) = 0 : 0 e^0 = - (0) + C 0 = -1 + C C = 1 Thus, the solution is y(x) = e^x(1 - x) . Since 1 - x 0 for all x , we have y(x) 0 on the interval [0, ] . The required area is: A = ₀^ y(x) dx