Solve the differential equation: dy/dx = (x + y) / x, with y(1) = 1.
Correct: A
dy/dx = 1 + y/x. Let v = y/x, so y = vx and dy/dx = v + x dv/dx. Then v + x dv/dx = 1 + v, so x dv/dx = 1. dv = dx/x, so v = ln|x| + C. Therefore, y/x = ln|x| + C, so y = x ln|x| + Cx. Using y(1) = 1, 1 = 1 ln|1| + C(1) => 1 = 0 + C, so C = 1. Thus, y = x ln|x| + x.