The solution of the differential equation d y d x +y x= 1 2 2 x

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + Py = Q , where P = x and Q = 1 2 2x = x x . The integrating factor (IF) is given by IF = e^ P dx = e^ x dx = e^ x . The solution of the differential equation is y IF = (Q IF) dx + c . Substituting the values, we get y e^ x = ( x x e^ x ) dx + c . Let u = x , then du = x dx . The integral becomes u e^u du . Using integration by parts, u e^u du = u e^u - e^u du = u e^u - e^u = e^u(u - 1) . Substituting back u = x , we get y e^ x = e