Formation of Differential Equations

Let f (x, y, c1) = 0 be an equation containing x, y and one arbitrary constant c1. If c1 is eliminated by differentiating f (x, y, c1) = 0 with respect to the independent variable once, you get a relation involving x, y and dy/dx, which is a differential equation of the first order.

Similarly, if you have an equation f(x, y, c1, c2) = 0 containing two arbitrary constants c1 and c2, then by differentiating this twice, you get three equations (including f). If the two arbitrary constants c1 and c2 are eliminated from these equations, you get a differential equation of second order.