Chapter XIII: Part 13

where A1, A2, ... An are arbitrary constants, and u is any particular
solution whatever; but if there be one root [t]1 repeated r times, the
terms A1 e^[t]1^x + ... + A_r e^[t]_r^x must be replaced by (A1 + A2x
+ ... + A_r x^r-1)e^[t]1x where A1, ... An are arbitrary constants;
the remaining terms in the complementary function will similarly need
alteration of form if there be other repeated roots.

To complete the solution of the differential equation we need some
method of determining a particular integral u; we explain a procedure
which is effective for this purpose in the cases in which R is a sum
of terms of the form e^ax[p](x), where [p](x) is an integral
polynomial in x; this includes cases in which R contains terms of the
form cos bx.[p](x) or sin bx.[p](x). Denote d/dx by D; it is clear
that if u be any function of x, D(e^ax u) = e^ax Du + ae^ax u, or say,
D(e^ax u) = e^ax (D + a)u; hence D^2(e^ax u), i.e. d^2/dx^2 (e^ax u),
being equal to D(e^ax v), where v=(D + a)u, is equal to e^ax(D + a)v,
that is to e^ax(D + a)^2u. In this way we find D^n(e^ax u) = e^ax(D +
a)^n u, where n is any positive integer. Hence if [psi](D) be any
polynomial in D with constant coefficients, [psi](D)(e^ax u) = e^ax
[psi](D + a)u. Next, denoting [int] udx by D^-1 u, and any solution of
the differential equation dz/dx + az = u by z = (d + a)^-1 u, we have
D[e^ax(D + a)^-1 u] = D(e^ax z) = e^ax(D + a)z = e^ax u, so that we
may write D^-1(e^ax u) = e^ax(D+a)^-1 u, where the meaning is that one
value of the left side is equal to one value of the right side; from
this, the expression D^-2(e^axu), which means D^-1[D^-1(e^ax u)], is
equal to D^-1(e^ax z) and hence to e^ax(D + a)^-1 z, which we write
e^ax(D + a)^-2 u; proceeding thus we obtain

D^-n(e^ax u) = e^ax(D + a)^-n u,

where n is any positive integer, and the meaning, as before, is that
one value of the first expression is equal to one value of the second.
More generally, if [psi](D) be any polynomial in D with constant
coefficients, and we agree to denote by 1/[psi](D) u any solution z of
the differential equation [psi](D)z = u, we have, if v = 1/[psi](D +
a) u, the identity [psi](D)(e^ax v) = e^ax [psi](D + a)v = e^ax u,
which we write in the form

1 1
--------(e^ax u) = e^ax ------------ u.
[psi](D) [psi](D + a)

This gives us the first step in the method we are explaining, namely
that a solution of the differential equation [psi](D)y = e^ax u + e^bx
v + ... where u, v, ... are any functions of x, is any function
denoted by the expression

1 1
e^ax ------------ u + e^ax ------------ v + ....
[psi](D + a) [psi](D + b)

It is now to be shown how to obtain one value of 1/[psi](D + a) u,
when u is a polynomial in x, namely one solution of the differential
equation [psi](D + a)z = u. Let the highest power of x entering in u
be x^m; if t were a variable quantity, the rational fraction in t,
1/[psi](t + a), by first writing it as a sum of partial fractions, or
otherwise, could be identically written in the form

K_r t^-r + K_r-1 t^-r+1 + ... + K1 t^-1 + H + H1t + ... + H_m t^m + t^m+1 [p](t)/[psi](t + a),

where [p](t) is a polynomial in t; this shows that there exists an
identity of the form

1 = [psi](t + a)(K_r t^-r + ... + K1t^-1 + H + H1t + ... + H_m t^m) + [p](t)t^m+1,

and hence an identity

u = [psi](D + a)[K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m]u + [p](D)D^m+1 u;

in this, since u contains no power of x higher than x^m, the second
term on the right may be omitted. We thus reach the conclusion that a
solution of the differential equation [psi](D + a)z = u is given by

z = (K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m)u,

of which the operator on the right is obtained simply by expanding
1/[psi](D + a) in ascending powers of D, as if D were a numerical
quantity, the expansion being carried as far as the highest power of D
which, operating upon u, does not give zero. In this form every term
in z is capable of immediate calculation.

_Example._--For the equation

d^4v d^2y
---- + 2--- + y = x^3 cos x or (D^2 + 1)^2y = x^3 cos x,
dx^4 dx^3

the roots of the associated algebraic equation ([t]^2+1)^2 = 0 are [t] =
[+-]i, each repeated; the complementary function is thus

(A + Bx)e^ix + (C + Dx)e^ix,

where A, B, C, D are arbitrary constants; this is the same as

(H + Kx) cos x + (M + Nx) sin x,

where H, K, M, N are arbitrary constants. To obtain a particular
integral we must find a value of (1 + D^2)^-2 x^3 cos x; this is the
real part of (1 + D^2)^-2 e^ix x^3 and hence of e^ix [1 + (D + i)^2]^-2
x^3

or e^ix [2iD(1 + 1/2iD)]^-2 x^3,

or -1/4e^ix D^-2 (1 + iD - 3/4D^2 - 1/2iD^3 + 5/16 D^4 + 3/16 iD^5 ...)x^3,

or -1/4e^ix(1/20 x^5 + 1/4ix^4 - 3/4x^3 - 3/2 ix^2 + 15/8 x + 9/8 i);

the real part of this is

-1/4(1/20 x^5 - 3/4x^2 + 15/8 x) cos x + 1/4(1/4x^4 - 3/2 x^2 + 9/8) sin x.

This expression added to the complementary function found above gives
the complete integral; and no generality is lost by omitting from the
particular integral the terms -15/32 x cos x + 9/32 sin x, which are
of the types of terms already occurring in the complementary function.

The symbolical method which has been explained has wider applications
than that to which we have, for simplicity of explanation, restricted
it. For example, if [psi](x) be any function of x, and a1, a2, ... an
be different constants, and [(t + a1) (t + a2) ... (t + an)]^-1 when
expressed in partial fractions be written [Sigma]c_m(t + a_m)^-1, a
particular integral of the differential equation (D + a1)(D + a2) ...
(D + a_n)y = [psi](x) is given by

y = [Sigma]c_m(D + a_m)^-1 [psi](x) = [Sigma]c_m(D + a_m)^-1 e^-a m^x e^a m^x [psi](x) =

[Sigma]c_m e^-a m^x D^-1 (e^a m^x [psi](x)) = [Sigma]c_m e^-a m^x [int] e^a m^x [psi](x)dx.

The particular integral is thus expressed as a sum of n integrals.

A linear differential equation of which the left side has the form

d^ny d^n-1 y dy
x^n ---- + P1x^n-1 ------- + ... + P_n-1 x-- + P_n y,
dx^n dx^n-1 dx

where P1, ... Pn are constants, can be reduced to the case considered
above. Writing x = e^t we have the identity

d^mu
x^m ---- = [t]([t] - 1)([t] - 2) ... ([t] - m + 1)u, where [t] = d/dt.
dx^m

When the linear differential equation, which we take to be of the
second order, has variable coefficients, though there is no general
rule for obtaining a solution in finite terms, there are some results
which it is of advantage to have in mind. We have seen that if one
solution of the equation obtained by putting the right side zero, say
y1, be known, the equation can be solved. If y2 be another solution of

d^2y dy
---- + P-- + Qy = 0,
dx^2 dx

there being no relation of the form my1 + ny2 = k, where m, n, k are
constants, it is easy to see that

d/dx(y1'y2 - y1y2') = P(y1'y2 - y1y2'),

so that we have

y1'y2 - y1y2' = A exp.([int] Pdx),

where A is a suitably chosen constant, and exp. z denotes e^z. In
terms of the two solutions y1, y2 of the differential equation having
zero on the right side, the general solution of the equation with R =
[phi](x) on the right side can at once be verified to be Ay1 + By2 +
y1u - y2v, where u, v respectively denote the integrals
_ _
/ /
u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx.
_/ _/

The equation

d^2y dy
---- + P-- + Qy = 0,
dx^2 dx

by writing y = v exp. (-1/2 [int] Pdx), is at once seen to be reduced to
d^2v/dx^2 + 1v = 0, where 1 = Q - 1/2dP/dx - 1/4P^2. If [eta] = - 1/v dv/dx,
the equation d^2v/dx^2 + 1v = 0 becomes d[eta]/dx = 1 + [eta]^2, a
non-linear equation of the first order.

More generally the equation

d[eta]
------ = A + B[eta] + C[eta]^2,
dx

where A, B, C are functions of x, is, by the substitution

1 dy
[eta] = - -- --,
Cy dx

reduced to the linear equation

d^2y / 1 dC\ dy
---- - ( B + - -- )-- + ACy = 0.
dx^2 \ C dx/ dx

The equation

d[eta]
------ = A + B[eta] + C[eta]^2,
dx

known as Riccati's equation, is transformed into an equation of the
same form by a substitution of the form [eta] = (aY + b)/(cY + d),
where a, b, c, d are any functions of x, and this fact may be utilized
to obtain a solution when A, B, C have special forms; in particular if
any particular solution of the equation be known, say [eta]0, the
substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the
general solution; for instance, when

d /A\
2B = -- log( - ),
dx \C/

a particular solution is [eta]0 = [root](-A/C). This is a case of the
remark, often useful in practice, that the linear equation

d^2y d[phi] dy
[phi](x)---- + 1/2------ -- + [mu]y = 0,
dx^2 dx dx

where [mu] is a constant, is reducible to a standard form by taking a
new independent variable
_
/
z = | dx[[p](x)]^-1/2.
_/

We pass to other types of equations of which the solution can be
obtained by rule. We may have cases in which there are two dependent
variables, x and y, and one independent variable t, the differential
coefficients dx/dt, dy/dt being given as functions of x, y and t. Of
such equations a simple case is expressed by the pair

dx dy
-- = ax + by + c, -- = a'x + b'y + c',
dt dt

wherein the coefficients a, b, c, a', b', c', are constants. To
integrate these, form with the constant [lambda] the differential
coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x +
(b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so
chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have
dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at
once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a
+ [lambda]a')t, where A is an arbitrary constant. In general, the
condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by
two different values of [lambda], say [lambda]1, [lambda]2; the
solutions corresponding to these give the values of x +[lambda]1y and
x + [lambda]2y, from which x and y can be found as functions of t,
involving two arbitrary constants. If, however, the two roots of the
quadratic equation for [lambda] are equal, that is, if (a - b')^2 +
4a'b = 0, the method described gives only one equation, expressing x +
[lambda]y in terms of t; by means of this equation y can be eliminated
from dx/dt = ax + by + c, leading to an equation of the form dx/dt =
Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The
integration of this gives x, and thence y can be found.

A similar process is applicable when we have three or more dependent
variables whose differential coefficients in regard to the single
independent variables are given as linear functions of the dependent
variables with constant coefficients.

Another method of solution of the equations

dx/dt = ax + by + c, dy/dt = a'x + b'y + c',

consists in differentiating the first equation, thereby obtaining

d^2x dx dy
---- = a-- + b--;
dt^2 dt dx

from the two given equations, by elimination of y, we can express
dy/dt as a linear function of x and dx/dt; we can thus form an
equation of the shape d^2x/dt^2 = P + Qx + Rdx/dt, where P, Q, R are
constants; this can be integrated by methods previously explained, and
the integral, involving two arbitrary constants, gives, by the
equation dx/dt = ax + by + c, the corresponding value of y. Conversely
it should be noticed that any single linear differential equation

d^2x dx
---- = u + vx + w--,
dt^2 dt

where u, v, w are functions of t, by writing y for dx/dt, is
equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In
fact a similar reduction is possible for any system of differential
equations with one independent variable.

Equations occur to be integrated of the form

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)!

Now compare with the given equation another equation

A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,

wherein each coefficient in the expansion of either A or B is real and
positive, and not less than the absolute value of the corresponding
coefficient in the expansion of a or b. In the second equation let us
substitute a series

F = P0 + tP1 + t^2P2/2! + ...,

wherein the coefficients in P0 are real and positive, and each not
less than the absolute value of the corresponding coefficient in p0;
then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary
equations of the same form as before, namely,

P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...

and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
[Delta]_s P0. These give for every coefficient in Ps+1 an integral
aggregate with real positive coefficients of the coefficients in P_s,
P_s-1, ..., P0 and the coefficients in A and B; and they are the same
aggregates as would be given by the previously obtained equations for
the corresponding coefficients in p_s+1 in terms of the coefficients
in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
coefficients in P0 and also in A, B are real and positive, it follows
that the values obtained in succession for the coefficients in P1, P2,
... are real and positive; and further, taking account of the fact
that the absolute value of a sum of terms is not greater than the sum
of the absolute values of the terms, it follows, for each value of s,
that every coefficient in p_s+1 is, in absolute value, not greater
than the corresponding coefficient in P_s+1. Thus if the series for F
be convergent, the series for f will also be; and we are thus reduced
to (1), specifying functions A, B with real positive coefficients,
each in absolute value not less than the corresponding coefficient in
a, b; (2) proving that the equation

AdF/dx + BdF/dy = dF/dt

possesses an integral P0 + tP1 + t^2P2/2! + ... in which the
coefficients in P0 are real and positive, and each not less than the
absolute value of the corresponding coefficient in p0. If a, b be
developable for x, y both in absolute value less than r and for t less
in absolute value than R, and for such values a, b be both less in
absolute value than the real positive constant M, it is not difficult
to verify that we may take

/ x + y\-1 / t\-1
A = B = M( 1 - ----- ) ( 1 - - ),
\ r / \ R/

and obtain
_ _
| 4MR / x + y\-2 / t\-1 |1/2
F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - ) |,
|_ r \ r / \ R/ _|

and that this solves the problem when x, y, t are sufficiently small
for the two cases p0 = x, p0 = y. One obvious application of the
general theorem is to the proof of the existence of an integral of an
ordinary linear differential equation given by the n equations dy/dx =
y1, dy1/dx = y2, ...,

dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;

but in fact any simultaneous system of ordinary equations is reducible
to a system of the form

dx1/dt = [phi](tx1, ... x_n).

Simultaneous linear partial equations.

Complete systems of linear partial equations.

Jacobian systems.

Suppose we have k homogeneous linear partial equations of the first
order in n independent variables, the general equation being
a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
... k, and that we desire to know whether the equations have common
solutions, and if so, how many. It is to be understood that the
equations are linearly independent, which implies that k <= n and not
every determinant of k rows and columns is identically zero in the
matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
= 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
[sigma]-th equation by P[sigma]f, it is clear that every common
solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
immediately find, however, that this is also a linear equation,
namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
P[sigma]a[rho]_i, and if it be not already contained among the given
equations, or be linearly deducible from them, it may be added to
them, as not introducing any additional limitation of the possibility
of their having common solutions. Proceeding thus with every pair of
the original equations, and then with every pair of the possibly
augmented system so obtained, and so on continually, we shall arrive
at a system of equations, linearly independent of each other and
therefore not more than n in number, such that the combination, in the
way described, of every pair of them, leads to an equation which is
linearly deducible from them. If the number of this so-called
_complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
leading to the nugatory result f = a constant. Suppose, then, the
number of this system to be r < n; suppose, further, that from the
matrix of the coefficients a determinant of r rows and columns not
vanishing identically is that formed by the coefficients of the
differential coefficients of f in regard to x1 ... x_r; also that the
coefficients are all developable about the values x1 = x1^0, ... xn=
xn^0, and that for these values the determinant just spoken of is not
zero. Then the main theorem is that the complete system of r
equations, and therefore the originally given set of k equations,
have in common n - r solutions, say [omega]r+1, ... [omega]n, which
reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
respectively put x1^0, ... x_r^0; so that also the equations have in
common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
arbitrary function [psi](x_r+1, ... x_n) which is developable about
x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
... [omega]_n). It is seen at once that this result is a
generalization of the theorem for r = 1, and its proof is conveniently
given by induction from that case. It can be verified without
difficulty (1) that if from the r equations of the complete system we
form r independent linear aggregates, with coefficients not
necessarily constants, the new system is also a complete system; (2)
that if in place of the independent variables x1, ... xn we introduce
any other variables which are independent functions of the former, the
new equations also form a complete system. It is convenient, then,
from the complete system of r equations to form r new equations by
solving separately for df/dx1, ..., df/dx_r; suppose the general
equation of the new system to be

Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).

Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
... Qrf, it must be identically zero. So reduced the system is called
a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
reducing respectively to x2, ... xn when

x1 = x1^0,

and its form shows that of these the first r - 1 are exactly x2 ...
xr. Let these n - 1 functions together with x1 be introduced as n new
independent variables in all the r equations. Since the first equation
is satisfied by n - 1 of the new independent variables, it will
contain no differential coefficients in regard to them, and will
reduce therefore simply to df/dx1 = 0, expressing that any common
solution of the r equations is a function only of the n - 1 remaining
variables. Thereby the investigation of the common solutions is
reduced to the same problem for r - 1 equations in n - 1 variables.
Proceeding thus, we reach at length one equation in n - r + 1
variables, from which, by retracing the analysis, the proposition
stated is seen to follow.

System of total differential equations.

The analogy with the case of one equation is, however, still closer.
With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
That consistent with them we may be able to regard x_r+1, ... x_n as
functions of x1, ... x_r, these being regarded as independent
variables, it is clearly necessary that when we differentiate
c_[sigma]j in regard to x_[rho] on this hypothesis the result should
be the same as when we differentiate c[rho]j, in regard to x[sigma] on
this hypothesis. The differential coefficient of a function f of x1,
... xn on this hypothesis, in regard to x_[rho]j is, however,

df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,

namely, is Q_[rho]f. Thus the consistence of the n - r total equations
requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
[omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
= x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
[omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
[psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
... x_r = x_r^0. And the total equations have no other solutions with
these initial values. Conversely, the existence of these solutions of
the total equations can be deduced a priori and the theory of the
Jacobian system based upon them. The theory of such total equations,
in general, finds its natural place under the heading _Pfaffian
Expressions_, below.

Geometrical interpretation and solution.

Mayer's method of integration.

A practical method of reducing the solution of the r equations of a
Jacobian system to that of a single equation in n - r + 1 variables
may be explained in connexion with a geometrical interpretation which
will perhaps be clearer in a particular case, say n = 3, r = 2. There
is then only one total equation, say dz = adz + bdy; if we do not take
account of the condition of integrability, which is in this case da/dy
+ bda/dz = db/dx + adb/dz, this equation may be regarded as defining
through an arbitrary point (x0, y0, z0) of three-dimensioned space
(about which a, b are developable) a plane, namely, z - z0 = a0(x -
x0) + b0(y - y0), and therefore, through this arbitrary point [oo]^2
directions, namely, all those in the plane. If now there be a surface
z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
y0, z0), this plane will touch the surface, and the operations of
passing along the surface from (x0, y0, z0) to

(x0 + dx0, y0, z0 + dz0)

and then to (x0 + dx0, y0 + dy0, Z0 + d^1z0), ought to lead to the same
value of d^1z0 as do the operations of passing along the surface from
(x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to

(x_ + dx_ , y_ + dy_ , Z_ + [delta]^1z_ ),
0 0 0 0 0 0

namely, [delta]^1z0 ought to be equal to d^1z0. But we find

d^1z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =

/db db \
a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
\dx0 dz0/

and so at once reach the condition of integrability. If now we put x
= x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be
considering the section of the surface by a fixed plane y - y0 = m(x -
x0); along this section dz = dt(a + bm); if we then integrate the
equation dx/dt = a + bm, where a, b are expressed as functions of m
and t, with m kept constant, finding the solution which reduces to z0
for t = 0, and in the result again replace m by (y - y0)/(x - x0), we
shall have the surface in question. In the general case the equations

dx_j - c_1j dx1 + ... c_rj dx_r

similarly determine through an arbitrary point x1^0, ... xn^0 a planar
manifold of r dimensions in space of n dimensions, and when the
conditions of integrability are satisfied, every direction in this
manifold through this point is tangent to the manifold of r
dimensions, expressed by [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
which satisfies the equations and passes through this point. If we put
x1 = x1^0 = t, x2 = x2^0 = m2t, ... xr = xr^0 = mrt, and regard m2,
... mr as fixed, the (n-r) total equations take the form dx_j/dt =
c_1j + m2c_2j + ... + m_rc_rj, and their integration is equivalent to
that of the single partial equation

n
df/dt + [Sigma](c_1j + m2c_2j + ... + m_rc_rj)df/dx_j = 0
j=r+1

in the n - r + 1 variables t, xr+1, ... xn. Determining the solutions
[Omega]_r+1, ... [Omega]_n which reduce to respectively x_r+1, ... x_n
when t = 0, and substituting t = x1 - x1^0, m2 = (x2 - x2^0)/(x1 -
x1^0), ... mr = (xr - xr^0)/(x1 - x1^0), we obtain the solutions of
the original system of partial equations previously denoted by
[omega]_r+1, ... [omega]_n. It is to be remarked, however, that the
presence of the fixed parameters m2, ... mr in the single integration
may frequently render it more difficult than if they were assigned
numerical quantities.

Pfaffian Expressions.

We have above considered the integration of an equation

dz = adz + bdy

on the hypothesis that the condition

da/dy + bda/dz = db/dz + adb/dz.

It is natural to inquire what relations among x, y, z, if any, are
implied by, or are consistent with, a differential relation adx + bdy
+ cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
problem leads to the consideration of the so-called _Pfaffian
Expression_ adx + bdy + cdz. It can be shown (1) if each of the
quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
denote respectively by u23, u31, u12, be identically zero, the
expression is the differential of a function of x, y, z, equal to dt
say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
the expression is of the form udt, i.e. it can be made a perfect
differential by multiplication by the factor 1/u; (3) that in general
the expression is of the form dt + u1dt1. Consider the matrix of four
rows and three columns, in which the elements of the first row are a,
b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
easily seen that the cases (1), (2), (3) above correspond respectively
to the cases when (1) every determinant of this matrix of two rows and
columns is zero, (2) every determinant of three rows and columns is
zero, (3) when no condition is assumed. This result can be generalized
as follows: if a1, ... an be any functions of x1, ... xn, the
so-called Pfaffian expression a1dx1 + ... + a_ndx_n can be reduced to
one or other of the two forms

u1dt1 + ... + u_kdt_k, dt + u1dt1 + ... + u_k-1 dt_k-1,

wherein t, u1 ..., t1, ... are independent functions of x1, ... xn,
and k is such that in these two cases respectively 2k or 2k - 1 is the
rank of a certain matrix of n + 1 rows and n columns, that is, the
greatest number of rows and columns in a non-vanishing determinant of
the matrix; the matrix is that whose first row is constituted by the
quantities a1, ... an, whose s-th element in the (r+1)-th row is the
quantity da_r/dx_s - da_s/dx_r. The proof of such a reduced form can
be obtained from the two results: (1) If t be any given function of
the 2m independent variables u1, ... um, t1, ... tm, the expression dt
+ u1 dt1 + ... + u_m dt_m can be put into the form u'1 dt'1 + ... +
u'_mdt'_m. (2) If the quantities u1, ..., u1, t1, ... tm be connected
by a relation, the expression n1dt1 + ... + umdtm can be put into the
format dt' + u'1 dt'1 + ... + u'_m-1 dt'_m-1; and if the relation
connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t'
can be taken to be zero. These two results are deductions from the
theory of _contact transformations_ (see below), and their
demonstration requires, beside elementary algebraical considerations,
only the theory of complete systems of linear homogeneous partial
differential equations of the first order. When the existence of the
reduced form of the Pfaffian expression containing only independent
quantities is thus once assured, the identification of the number k
with that defined by the specified matrix may, with some difficulty,
be made _a posteriori_.

Single linear Pfaffian equation.

In all cases of a single Pfaffian equation we are thus led to consider
what is implied by a relation dt - u1dt1 - ... - umdtm = 0, in which
t, u1, ... um, t1 ..., tm are, except for this equation, independent
variables. This is to be satisfied in virtue of one or several
relations connecting the variables; these must involve relations
connecting t, t1, ... tm only, and in one of these at least t must
actually enter. We can then suppose that in one actual system of
relations in virtue of which the Pfaffian equation is satisfied, all
the relations connecting t, t1 ... tm only are given by

t = [psi](t_s+1 ... t_m), t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);

so that the equation

d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0

is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
to zero the coefficients of the differentials of these variables, we
thus obtain m - s relations of the form

d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;

these m - s relations, with the previous s + 1 relations, constitute a
set of m + 1 relations connecting the 2m + 1 variables in virtue of
which the Pfaffian equation is satisfied independently of the form of
the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
any value of s there may exist relations additional to the specified m
+ 1 relations, provided they do not involve any relation connecting t,
t1, ... tm only, and are consistent with the m - s relations
connecting u1, ... um. It is now evident that, essentially, the
integration of a Pfaffian equation

a1dx1 + ... + a_n dx_n = 0,

wherein a1, ... an are functions of x1, ... xn, is effected by the
processes necessary to bring it to its reduced form, involving only
independent variables. And it is easy to see that if we suppose this
reduction to be carried out in all possible ways, there is no need to
distinguish the classes of integrals corresponding to the various
values of s; for it can be verified without difficulty that by putting
t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
... - u'_m dt'_m = 0, and the general relations changed to

t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) - ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],

say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
contain only one relation connecting the variables t', t'1, ... t'm
only.

Simultaneous Pfaffian equations.

This method for a single Pfaffian equation can, strictly speaking, be
generalized to a simultaneous system of (n - r) Pfaffian equations dxj
= c_1j dx1 + ... + c_rj dxr only in the case already treated, when
this system is satisfied by regarding x_r+1, ... x_n as suitable
functions of the independent variables x1, ... xr; in that case the
integral manifolds are of r dimensions. When these are non-existent,
there may be integral manifolds of higher dimensions; for if

d[phi] = [phi]1 dx_r + ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 + ... + c_r,r+1 dx_r) + [phi]_r+2 ( ) + ...

be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
+ c[sigma]_,n [phi]_n = 0, or [phi] satisfies the r partial
differential equations previously associated with the total equations;
when these are not a complete system, but included in a complete
system of r - [mu] equations, having therefore n - r - [mu]
independent integrals, the total equations are satisfied over a
manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
(1901), p. 386).

Contact transformations.

It seems desirable to add here certain results, largely of algebraic
character, which naturally arise in connexion with the theory of
contact transformations. For any two functions of the 2n independent
variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
... pn we denote by [phi][psi] the sum of the n terms such as

d[phi] /d[psi] d[psi]\ d[psi] /d[phi] d[phi]\
------( ------ + p_i------ ) - ------( ------ + p_i------ ).
dpi \ dxi dz / dpi \ dxi dz /