eoretically, be reduced to the type dy/dx = F(x); similarly an
equation F(y, dy/dx) = 0.

(3) An equation

f(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 [phi]1(x, y, c) = 0, where c is an
arbitrary constant; the product equation [phi]1(x, y, c)[phi]2(x, y,
c) ... = 0, consisting of as many factors as there were values of
dy/dx, is effectively as general as if we wrote [phi]1(x, y, c1)
[phi]2(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 [phi]1(x, y, c)[phi]2(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 + Cy^2

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 d^2y/dx^2,
the number of types for which the solution can be found by a known
procedure is very considerably reduced. Consider the general linear
equation

d^2y dy
--- + P-- + Qy = R,
dx^2 dx

where P, Q, R are functions of x only. There is no method always
effective; the main general result for such a linear equation is that
if any particular function of x, say y1, can be discovered, for which

d^2y1 dy1
---- + P--- + Qy1 = 0,
dx^2 dx

then the substitution y = y1[eta] in the original equation, with R on
the right side, reduces this to a linear equation of the first order
with the dependent variable d[eta]/dx. In fact, if y = y1[eta] we have

dy d[eta] dy1 d^2y d^2[eta] dy1 d[eta]