Xdx + Ydy + Zdz = 0,
where X, Y, Z are functions of x, y, z. We consider only the case in
which there exists an equation [phi](x, y, z) = C whose differential
dP[phi] dP[phi] dP[phi]
-------dx + -------dy + -------dz = 0
dPx dPy dPz
is equivalent with the given differential equation; that is, [mu]
being a proper function of x, y, z, we assume that there exist
equations
dP[phi] dP[phi] v[phi]
------- = [mu]X, ------- = [mu]Y, ------ = [mu]Z;
dPx vy vz
these equations require
dP dP
---([mu]Y) = ---([mu]Z), &c.,
dPz dPy
and hence
/dPZ dPY\ /dPX dPZ\ /dPY dPX\
X( --- - --- ) + Y( --- - --- ) + Z( --- - --- ) = 0;
\dPy dPz/ \dPz dPx/ \dPx dPy/
conversely it can be proved that this is sufficient in order that [mu]
may exist to render [mu](Xdx + Ydy + Zdz) a perfect differential; in
particular it may be satisfied in virtue of the three equations such
as
dPZ dPY
--- - --- = 0;
dPy dPz
in which case we may take [mu] = 1. Assuming the condition in its
general form, take in the given differential equation a plane section
of the surface [phi] = C parallel to the plane z, viz. put z constant,
and consider the resulting differential equation in the two variables
x, y, namely Xdx + Ydy = 0; let [psi](x, y, z) = constant, be its
integral, the constant z entering, as a rule, in [psi] because it
enters in X and Y. Now differentiate the relation [psi](x, y, z) =
[f](z), where [f] is a function to be determined, so obtaining
dP[psi] dP[psi] /dP[psi] df\
-------dx + -------dy + ( ------- - -- )dz = 0;
dPx dPy \ dPz dz/
there exists a function [sigma] of x, y, z such that
dP[psi] dP[psi]
-------- = [sigma]X, ------- = [sigma]Y,
dPx dPy
because [psi] = constant, is the integral of Xdx + Ydy = 0; we desire
to prove that [f] can be chosen so that also, in virtue of [psi](x, y,
z) = f(z), we have
dP[psi] df df dP[psi]
------- - -- = [sigma]Z, namely -- = ------- - [sigma]Z;
dPz dz dz dPz
if this can be proved the relation [psi](x, y, z) - f(z) = constant,
will be the integral of the given differential equation. To prove this
it is enough to show that, in virtue of [psi](x, y, z) = [f](z), the
function dP[psi]/dPx - [sigma]Z can be expressed in terms of z only.
Now in consequence of the originally assumed relations,
dP[psi] dP[phi] dP[phi]
------- = [mu]X, ------- = [mu]Y, ------- = [mu]Z,
dPx dPy dPz
we have
dP[psi] /dP[phi] [sigma] dP[psi] /dP[phi]
------- / ------- = ------- = ------- / -------,
dPx / dPx [mu] dPy / dPy
and hence
dP[psi] dP[phi] dP[psi] dP[phi]
------- ------- - ------- ------- = 0;
dPx dPy dPy dPx
this shows that, as functions of x and y, [psi] is a function of [phi]
(see the note at the end of part i. of this article, on Jacobian
determinants), so that we may write [psi] = F(z, [phi]), from which
[sigma] dPF dP[psi] dPF dPF dP[phi] dPF [sigma] dPF
------- = -------; then ------- = --- + ------- ------- = --- + ------- . [mu]Z = --- + [sigma]Z
[mu] dP[phi] dPz dPz dP[phi] dPz dPz [mu] dPz
dP[psi] dPF
or ------- - [sigma]Z = ---;
dPz dPz
in virtue of [psi](x, y, z) = f(z), and [psi] = F(z, [phi]), the
function [phi] can be written in terms of z only, thus dPF/dPz can be
written in terms of z only, and what we required to prove is proved.
Consider lastly a simple type of differential equation containing
_two_ independent variables, say x and y, and one dependent variable
z, namely the equation
dPz dPz
P--- + Q--- = R,
dPx dPy
where P, Q, R are functions of x, y, z. This is known as Lagrange's
linear partial differential equation of the first order. To integrate
this, consider first the ordinary differential equations dx/dz = P/R,
dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be
determined, independent of one another, such that the equations u = a,
v = b, where a, b are arbitrary constants, lead to these ordinary
differential equations, namely such that
dPu dPu dPu dPv dPv dPv
P--- + Q--- = R--- = 0 and P--- + Q--- = R--- = 0.
dPx dPy dPz dPx dPy dPz
Then if F(x, y, z) = 0 be a relation satisfying the original
differential equations, this relation giving rise to
dPF dPF dPz dPF dPF dPz dPF dPF dPF
--- + --- --- = 0 and --- + --- --- = 0, we have P--- + Q--- = R--- = 0.
dPx dPz dPx dPy dPz dPy dPx dPy dPz
It follows that the determinant of three rows and columns vanishes
whose first row consists of the three quantities dPF/dPx, dPF/dPy,
dPF/dPz, whose second row consists of the three quantities dPu/dPx,
dPu/dPy, dPu/dPz, whose third row consists similarly of the partial
derivatives of v. The vanishing of this so-called Jacobian determinant
is known to imply that F is expressible as a function of u and v,
unless these are themselves functionally related, which is contrary to
hypothesis (see the note below on Jacobian determinants). Conversely,
any relation [phi](u, v) = 0 can easily be proved, in virtue of the
equations satisfied by u and v, to lead to
dz dz
P-- + Q-- = R.
dx dx
The solution of this partial equation is thus reduced to the solution
of the two ordinary differential equations expressed by dx/P = dy/Q =
dz/R. In regard to this problem one remark may be made which is often
of use in practice: when one equation u = a has been found to satisfy
the differential equations, we may utilize this to obtain the second
equation v = b; for instance, we may, by means of u = a, eliminate
z--when then from the resulting equations in x and y a relation v = b
has been found containing x and y and a, the substitution a = u will
give a relation involving x, y, z.
_Note on Jacobian Determinants._--The fact assumed above that the
vanishing of the Jacobian determinant whose elements are the partial
derivatives of three functions F, u, v, of three variables x, y, z,
involves that there exists a functional relation connecting the three
functions F, u, v, may be proved somewhat roughly as follows:--
The corresponding theorem is true for any number of variables.
Consider first the case of two functions p, q, of two variables x, y.
The function p, not being constant, must contain one of the variables,
say x; we can then suppose x expressed in terms of y and the function
p; thus the function q can be expressed in terms of y and the function
p, say q = Q(p, y). This is clear enough in the simplest cases which
arise, when the functions are rational. Hence we have
dPq dPQ dPp dPq dPQ dPp dPQ
--- = --- --- and --- = --- --- + ---;
dPx dPp dPx dPy dPp dPy dPy
these give
dPp dPq dPp dPq dPp dPQ
--- --- - --- --- = --- ---;
dPx dPy dPy dPx dPx dPy
by hypothesis dPp/dPx is not identically zero; therefore if the
Jacobian determinant of p and q in regard to x and y is zero
identically, so is dPQ/dPy, or Q does not contain y, so that q is
expressible as a function of p only. Conversely, such an expression
can be seen at once to make the Jacobian of p and q vanish
identically.
Passing now to the case of three variables, suppose that the Jacobian
determinant of the three functions F, u, v in regard to x, y, z is
identically zero. We prove that if u, v are not themselves
functionally connected, F is expressible as a function of u and v.
Suppose first that the minors of the elements of dPF/dPx, dPF/dPy,
dPF/dPz in the determinant are all identically zero, namely the three
determinants such as
dPu dPv dPu dPv
--- --- - --- ---;
dPy dPz dPz dPy
then by the case of two variables considered above there exist three
functional relations. [psi]1(u, v, x) = 0, [psi]2(u, v, y) = 0,
[psi]3(u, v, z) = 0, of which the first, for example, follows from the
vanishing of
dPu dPv dPu dPv
--- --- - --- ---.
dPy dPz dPz dPy
We cannot assume that x is absent from [psi]1, or y from [psi]2, or z
from [psi]3; but conversely we cannot simultaneously have x entering
in [psi]1, and y in [psi]2, and z in [psi]3, or else by elimination of
u and v from the three equations [psi]1 = 0, [psi]2 = 0, [psi]3 = 0,
we should find a necessary relation connecting the three independent
quantities x, y, z; which is absurd. Thus when the three minors of
dPF/dPx, dPF/dPy, dPF/dPz in the Jacobian determinant are all zero,
there exists a functional relation connecting u and v only. Suppose no
such relation to exist; we can then suppose, for example, that
dPu dPv dPu dPv
--- --- - --- ---
dPy dPz dPz dPy
is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we
can express y and z in terms of u, v, and x (the attempt to do this
could only fail by leading to a relation connecting u, v and x, and
the existence of such a relation would involve that the determinant
dPu dPv dPu dPv
--- --- - --- ---
dPy dPz dPz dPy
was zero), and so write F in the form F(x, y, z) = [Phi](u, v, x). We
then have
dPF dP[Phi] dPu dP[Phi] dPv dP[Phi] dPF dP[Phi] dPu dP[Phi] dPv dPF dP[Phi] dPu dP[Phi] dPv
--- = ------- --- + ------- --- + -------, --- = ------- --- + ------- ---, --- = ------- --- + ------- ---;
dPx dPu dPx dPv dPx dPx dPy dPu dPy dPv dPy dPz dPu dPz dPv dPz
thereby the Jacobian determinant of F, u, v is reduced to
dP[Phi] /dPu dPv dPu dPv\
-------( --- --- - --- --- );
dPx \dPy dPz dPz dPy/
by hypothesis the second factor of this does not vanish identically;
hence dP[Phi]/dPx = 0 identically, and [Phi] does not contain x; so
that F is expressible in terms of u, v only; as was to be proved.
_Part II.--General Theory._
Differential equations arise in the expression of the relations between
quantities by the elimination of details, either unknown or regarded as
unessential to the formulation of the relations in question. They give
rise, therefore, to the two closely connected problems of determining
what arrangement of details is consistent with them, and of developing,
apart from these details, the general properties expressed by them. Very
roughly, two methods of study can be distinguished, with the names
Transformation-theories, Function-theories; the former is concerned with
the reduction of the algebraical relations to the fewest and simplest
forms, eventually with the hope of obtaining explicit expressions of the
dependent variables in terms of the independent variables; the latter is
concerned with the determination of the general descriptive relations
among the quantities which are involved by the differential equations,
with as little use of algebraical calculations as may be possible. Under
the former heading we may, with the assumption of a few theorems
belonging to the latter, arrange the theory of partial differential
equations and Pfaff's problem, with their geometrical interpretations,
as at present developed, and the applications of Lie's theory of
transformation-groups to partial and to ordinary equations; under the
latter, the study of linear differential equations in the manner
initiated by Riemann, the applications of discontinuous groups, the
theory of the singularities of integrals, and the study of potential
equations with existence-theorems arising therefrom. In order to be
clear we shall enter into some detail in regard to partial differential
equations of the first order, both those which are linear in any number
of variables and those not linear in two independent variables, and also
in regard to the function-theory of linear differential equations of the
second order. Space renders impossible anything further than the
briefest account of many other matters; in particular, the theories of
partial equations of higher than the first order, the function-theory of
the singularities of ordinary equations not linear and the applications
to differential geometry, are taken account of only in the bibliography.
It is believed that on the whole the article will be more useful to the
reader than if explanations of method had been further curtailed to
include more facts.
When we speak of a function without qualification, it is to be
understood that in the immediate neighbourhood of a particular set x0,
y0, ... of values of the independent variables x, y, ... of the
function, at whatever point of the range of values for x, y, ... under
consideration x0, y0, ... may be chosen, the function can be expressed
as a series of positive integral powers of the differences x - x0, y
-y0, ..., convergent when these are sufficiently small (see FUNCTION:
Functions of Complex Variables). Without this condition, which we
express by saying that the function is developable about x0, y0, ...,
many results provisionally stated in the transformation theories would
be unmeaning or incorrect. If, then, we have a set of k functions, f1
... fk of n independent variables x1 ... xn, we say that they are
independent when n >= k and not every determinant of k rows and columns
vanishes of the matrix of k rows and n columns whose r-th row has the
constituents dfr/dx1, ... dfr/dxn; the justification being in the
theorem, which we assume, that if the determinant involving, for
instance, the first k columns be not zero for x1 = x1^0 ... xn = xn^0,
and the functions be developable about this point, then from the
equations f1 = c1, ... fk = ck we can express x1, ... xk by convergent
power series in the differences x_k+1 - x_k+1^0, ... x_n - x_n^0, and so
regard x1, ... xk as functions of the remaining variables. This we often
express by saying that the equations f1 = c1, ... fk = ck can be solved
for x1, ... xk. The explanation is given as a type of explanation often
understood in what follows.
Ordinary equations of the first order.
Single homogeneous partial equation of the first order.
Proof of the existence of integrals.
We may conveniently begin by stating the theorem: If each of the n
functions [phi]1, ... [phi]n of the (n + 1) variables x1, ... x_nt be
developable about the values x1^0, ... x_n^0t^0, the n differential
equations of the form dx1/dt = [phi]1(tx1, ... xn) are satisfied by
convergent power series
x_r = x_r^0 + (t - t^0 ) A_r1 + (t - t0 )^2A_r2 + ...
reducing respectively to x1^0, ... xn^0 when t = t^0; and the only
functions satisfying the equations and reducing respectively to x1^0,
... xn^0 when t = t^0, are those determined by continuation of these
series. If the result of solving these n equations for x1^0, ... xn^0
be written in the form [omega]1(x1, ... xnt) = x1^0, ... [omega]n(x1,
... xnt) = xn^0, it is at once evident that the differential equation
df/dt + [phi]1 df/dx1 + ... + [phi]n df/dxn = 0
possesses n integrals, namely, the functions [omega]1, ... [omega]n,
which are developable about the values (x1^0 ... xn^0t^0) and reduce
respectively to x1, ... xn when t = t^0. And in fact it has no other
integrals so reducing. Thus this equation also possesses a unique
integral reducing when t = t^0 to an arbitrary function [psi](x1, ...
xn), this integral being. [psi]([omega]1, ... [omega]n). Conversely
the existence of these _principal_ integrals [omega]1, ... [omega]n of
the partial equation establishes the existence of the specified
solutions of the ordinary equations dxi/dt = [phi]i. The following
sketch of the proof of the existence of these principal integrals for
the case n = 2 will show the character of more general investigations.
Put x for x - x^0, &c., and consider the equation a(xyt) df/dx +
b(xyt) df/dy = df/dt, wherein the functions a, b are developable about
x = 0, y = 0, t = 0; say
a(xyt) = a0 + ta1 + t^2a2/2! + ..., b(xyt) = b0 + tb1 + t^2b2/2! + ...,
so that
ad/dx + bd/dy = [delta]0 + t[delta]1 + 1/2t^2[delta]2 + ...,
where [delta] = a_r d/dx + b_r d/dy. In order that
f = p0 + tp1 + t^2p2/2! + ...
wherein p0, p1 ... are power series in x, y, should satisfy the
equation, it is necessary, as we find by equating like terms, that
p1 = [delta]0 p0, p2 = [delta]0 p1 + [delta]1 p0, &c.
and in general
p_s+1 = [delta]0 p_s + s1 [delta]1 p_s-1 + ... + [delta]_s p0,
where s_r = (s!)/(r!) (s - r)!