ƒ(x, dy/dx) = 0,
may, theoretically, be reduced to the type dy/dx = F(x); similarly an equation F(y, dy/dx) = 0.
(3) An equation
ƒ(dy/dx, x, y) = 0,
which is an integral polynomial in dy/dx, may, theoretically, be solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x, y) corresponds, suppose, a solution φ1(x, y, c) = 0, where c is an arbitrary constant; the product equation φ1(x, y, c)φ2(x, y, c) ... = 0, consisting of as many factors as there were values of dy/dx, is effectively as general as if we wrote φ1(x, y, c1)φ2(x, y, c2) ... = 0; for, to evaluate the first form, we must necessarily consider the factors separately, and nothing is then gained by the multiple notation for the various arbitrary constants. The equation φ1(x, y, c)φ2(x, y, c) ... = 0 is thus the solution of the given differential equation.
In all these cases there is, except for cases of singular solutions, one and only one arbitrary constant in the most general solution of the differential equation; that this must necessarily be so we may take as obvious, the differential equation being supposed to arise by elimination of this constant from the equation expressing its solution and the equation obtainable from this by differentiation in regard to x.
A further type of differential equation of the first order, of the form
dy/dx = A + By + Cy²
in which A, B, C are functions of x, will be briefly considered below under differential equations of the second order.
When we pass to ordinary differential equations of the second order, that is, those expressing a relation between x, y, dy/dx and d²y/dx², the number of types for which the solution can be found by a known procedure is very considerably reduced. Consider the general linear equation