Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.

Given y = mx where m is an arbitrary constant. Differentiating both sides, we get dy/dx = m. Substituting for dy/dx = m in y = mx as shown below: y = x(dy/dx) ⇒ x(dy/dx) - y = 0. Hence the differential equation is x(dy/dx) - y = 0.