DIFFERENTIAL EQUATION, in mathematics, a relation between one or more functions and their differential coefficients. The subject is treated here in two parts: (I) an elementary introduction dealing with the more commonly recognized types of differential equations which can be solved by rule; and (2) the general theory.
Part I.—Elementary Introduction.
Of equations involving only one independent variable, x (known as ordinary differential equations), and one dependent variable, y, and containing only the first differential coefficient dy/dx (and therefore said to be of the first order), the simplest form is that reducible to the type
dy/dx =f(x)/F(y),
leading to the result fF(y)dyff(x)dx=A, where A is an arbitrary constant; this result is said to solve the differential equation, the problem of evaluating the integrals belonging to the integral calculus.
II
.)uo
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J }
Xmi1
Jxn
the number of types for which the solution can be found by a known procedure is very considerably reduced. Consider the general linear equation
226
Another simple form is
dy/dx+yP =Q,
where P, Q are functions of x only; this is known as the linear equation, since it contains y and dy/dx only to the first degree. If fPdx=u, we clearly have
so that y=e"(fe''Qdx+A) solves the equation, and is the only possible solution, A being an arbitrary constant. The rule for the solution of the linear equation is thus to multiply the equation by e", where u=fPdx.
A third simple and important form is that denoted by
y=px+f(p),
where p is an abbreviation for dy/dx; this is known as Clairaut's form. By differentiation in regard to x it gives
pp+xd +f'(P)a ' where
(p) =ad
 f(P) ;
thus, either (i.) dpldx=o, that is, p is constant on the curve satisfying the differential equation, which curve is thus any one of the straight lines y=cxbf(c), where c is an arbitrary constant, or else, (ii.) x+f'(p) =o; if this latter hypothesis be taken,andpbeeliminated between x+f'(p) =o and y=px+f(p), a relation connecting x and y, not containing an arbitrary constant, will be found, which obviously represents the envelope of the straight lines y=cx}f (c).
In general if a differential equation . (x, y, dy/dx) =o be satisfied by any one of the curves F(x, y, c) =o,where c is an arbitrary constant, it is clear that the envelope of these curves, when existent, must also satisfy the differential equation; for this equation prescribes a relation connecting only the coordinates x, y and the differential coefficient dy/dx, and these three quantities are the same at any point of the envelope for the envelope and for the particular curve of the family which there touches the envelope. The relation expressing the equation of the envelope is called a singular solution of the differential equation, meaning an isolated solution, as not being one of a family of curves depending upon an arbitrary parameter.
An extended form of Clairaut's equation expressed by
Y=xF(p)+f(p)
may be similarly solved by first differentiating in regard to p, when it reduces to a linear equation of which x is the dependent and p the independent variable ; from the integral of this linear equation, and the original differential equation, the quantity p is then to be eliminated.
Other types of solvable differential equations of the first order are (i)
Mdy/dx = N,
where M, N are homogeneous polynomials in x and. y, of the same order; by putting v=y/x and eliminating y, the equation becomes of the first type considered above, in v and x. An equation (aBsbA)
(ax+by+c)dy/dx=Ax+By+C
may be reduced to this rule by first putting x+h, y+k for x and y, and determining h,k so that ah+bk+c=o, Ah+Bk+C=o.
(2) An equation in which y does not explicitly occur,
f(a, dy/dx) =o,
may, theoretically, be reduced to the type dy/dx=F(x); similarly an equation F(y, dy/dx)=o.
(3) An equation
f(dy/dx, x, y) =o,
which is an integral polynomial in dy/dx, may, theoretically, be solved for dy/dx, as an algebraic equation; to any root dy/dx=Fi(x,y) corresponds, suppose, a solution 01(x, y, c) =o, where c is an arbitrary constant; the product equation '1(x, y, c)c2(x,y,c) ... =o, consisting of as many factors as there were values of dy/dx, is effectively as general as if we wrote ¢i(x, y; ci)¢2(x, y, C2) . =o; 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 ¢i(x, y, c)(x, y, c) ... =o 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,d 2+Pdx+QY=R,
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 yi, can be discovered, for which
d+Pdyi+QYi dx °o' dx2
then the substitution y=yin in the original equation, with R on the right side, reduces this to a linear equation of the first order with the dependent variable dn/dx. In fact, if y=yin we have
dy _yidn+ndy~anddiy — 1d2z+2dYidn nd2 y'
ax  ax dx dx2 —y dx dx dx
and thus
ZY Y 2n do d2yi dyi
dx2 +P dx +Qy=Yidx2 + 1 2 dx +Pyi) dx + \dx2 +P dx +QYi) n;
if then
+P T +QY,=o,
and z denote dn/dx, the original differential equation becomes dz + 1 /2 ax dyi
yi dx +Py) z = R.
From this equation z can be found by the rule given above for the linear equation of the first order, and will involve one arbi
trary constant; thence y =yi it = yi f zdx+Ay', where A is another
arbitrary constant, will be the general solution of the original equation, and, as was to be expected, involves two arbitrary constants.
The case of most frequent occurrence is that in which the coefficients P, Q are constants; we consider this case in some detail. If 0 be a root of the quadratic equation 02+OP+Q = o, it can be at once seen that a particular integral of the differential equation with zero on the right side is yl =eox. Supposing first the roots of the quadratic equation to be different, and o to be theother root, so that +O = P, the auxiliary differential equation for z, referred to above,
becomes dx + (0 — z = Re ~, which leads to ze(o"0) `= B +f Re dx, where B is an arbitrary constant, and hence to y=Aees{eB~~Be(0B)sdx+e" f e(~B>z f Re dxdx,
or say to y=Aeox+Cemx+U, where A, C are arbitrary constants and U is a function of x, not present at all when R=o. If the quadratic equation 02+P0+Q=o has equal roots, ..so that 20=—P, the
auxiliary equation inzbecomes dz/dx = Rex, givingz = B + f Re'dx, where B is an arbitrary constant, and hence y = (A+Bx)eek+ems f f Re9''dxdx,
or, say, y= (Af Bx)eox1U, where A, B are arbitrary constants, and U is a function of x not present at all when R=o. The portion Aeox+Bee' or (A+Bx)eox of the solution, which is known as the complementary function, can clearly be written down at once by inspection of the given differential equation. The remaining portion U may, by taking the constants in the complementary function properly, be replaced by any particular solution whatever of the differential equation
d?v dv
dx2+Pdx+QY=R;
for if u be any particular solution, this has a form
is =Aoeox+Boer+U,
or a form
u = (Ao+Box)eox+U ;
thus the general solution can be written
(AAo)ex+(BBOO' +u, or {AAs+(BBo)xleox+u, where A—A0, B—Bo, like A, B. are arbitrary constants.
A similar result holds for a linear differential equation of any order, say
dx(ye") =e" (xfPY) =e"Q,
do n+Pidn i 1 + . +P"y = R, dx dx
where Pi, P2, . . . P,, are constants, and R is a function of x. If we form thealgebraic equation 0"+PiO"i+ ... +P.= o, and all the roots of this equation be different, say they are 01, 02, . 0,,, the general solution of the differential equation is
y=Aie ix+A2e 2x+ ... +Anednx+u,
where Ai, A2, . . . An are arbitrary constants, and u  is any
particular solution whatever; but if there be one root 91 re
peated r times, the terms Aleelx+ ... +Are rx must be replaced by
(Al+Az+ ... +Arxr1)eel where Ai, ... A,, are arbitrary con
stants; 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 pro
cedure which is effective for this purpose: in the cases in which R is
a sum of terms of the form e°xcl)(x), where ¢(x) is an integral poly
nomial in x; this includes cases in which R contains terms of the
form cos bx. (x) or sin bx.o(x). Denote d/dx by D; it is clear that if
u be any function of x, D(e°xu) =eGZDu+ae°xu, or say, D(e6zu) =
2
e°'(D+a)u; hence D2(e°xu), i.e.dx2(e°xu), being equal to D(e"v), where v=(D+a)u, is equal to a°x (D+a)v, that is to e°x(D+a)2u. In this way we find Da(e' u)=e°x(D+a)au, where n is any positive
integer. Hence if >'(D) be any polynomial in D with constant coefficients, ii(D) (e°xu)=a°N,(D+a)u. Next, denoting f udx by D 'u, and any solution of the differential equation
dx+az = u by
z=(D+a)we have D[e°x(D+a)'u]=D(eaxz) =e°x(D+a)z= e°xu, so that we may write Dl(e' u) =e°x(D+a)1u, where the meaning is that one value of the left side is equal to one value of the right side; from this, the expression D2(e4zu), which means D1fD1(e°xu)], is equal to D1(e°xz) and hence to e°x(D+a)'z, which we write e°x(D+a)2u; proceeding thus we obtain
Da(eaxu) =e62(D+a)an
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 ¢(D) be any polynomial in D with constant co
efficients, and we agree to denote by~(D)u any solution z of the
differential equation >G(D)z=u, we have, if v=~(D +a)u, the identity
#(D) (e°xv) =e°xtp(D+a)v=e°xu, which we write in the form
0(D)(eaxu) _eax#(D+a)u.
This gives us the first step in the method we are explaining, namely that a solution of the differential equation ¢(D)y=e°xu+ ebxv+ . . . where u, v, . . . are any functions of x, is any function denoted by the expression
I I
eax./(D+a)u+ebz4,(D+b)v+ ••••
It is now to be shown how to obtain one value of ,(D+a)u when u
is a polynomial in x, namely one solution of the differential equation tp(D+a)z=u. Let the highest power of x entering in u be xm; if t
were a variable quantity, the rational fraction in t, ~(trfa ' by first writing it as a sum of partial fractions, or otherwise, could be identically written in the form
Kit'+ Kait' +...+Kit–l+H+Hit+• +Hint' +tm+iO(t)Ab(t+a), where ¢(t) is a polynomial in t; this shows that there exists an identity of the form
I = G(t+a) (Krt''+ ... +Kit–i+H+Hit+ .. . +Hmtm) +(t)tm+i and hence an identity
u=t/(D+a)[KrD''+ ... +KiDi+H+H1D+ ... +HmDm]u
+'(D) D1' lu;
in this, since u contains no power of x higher than xm, the second term on the right may be omitted. We thus reach the conclusion that a solution of the differential equation 1'(D+a)z=u is given by
z= (K,D''+ ... +KiDi+H+H1D+ ... + HmDm)u,
of which the operator on the right is obtained simply by expanding I/ty(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
dxd +2dx
~+y=x3 cos x or (D2lI)2y=xs cos x,
the roots of the associated algebraic equation (02+r)2=o area= each repeated ; the complementary function is thus (A+Bx)eix+(C+Dx)ecx
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 (I +D2)''2x3 cos x; this is. the real
part of (I+D2)–2 e'xx3 and hence of e'x[I+(D+i)2]2x8
or e's[2iD(I  A 2x,,
or ie=xD2(I+iD1.D2aiD3+ D4+AiD3...)x3,
or 4etx(2isx5+iix44x32ix2+Vx+ii);
the real part of this is
.220 cos x+4(4x4axe+1) 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 H x cos x+B9 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 >'(x) be any function of x, and at, a2, ...as be different constants, and [(t+ai) (t+a2) ... (t+a„011 when expressed in partial fractions be written Zcm(t+am)1, a particular integral of the differential equation (D+ai)(D+a2) .. . (D+aa)y=>/'(x) is given by
y=S..cm(D+am)–1 1,&(x) =z1crh(D+dm)–le(~°'axeama,(x) = Ecme`6mxD–1 (ear"x+G(x) I =Zcme'6m1 J e°m1tb(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 day n da2y dy
xa I I'txaldxa i+... +Pnxxdx+Pay,
where Pi, . . . P. are constants, can be reduced to the case considered above. Writing x=et we have the identity
dmu
xmclxf°=a(9I)(02). ..(9m+r) u, where a=d/dt.
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
dx2+PdxI QY = o,
there being no relation of the form mylEny2=k, where m, n, k are constants, it is easy to see that
d
dx(Yi'Y2y1Y2) = P(Yi'Y2Y1Y2'),
so that we have yl'Y2Yiy2' =A exp. (f Pdx) ,
where A is a suitably chosen constant, and exp. z denotes es. In terms of the two solutions yi, Y2 of the differential equation having zero on the right side, the general solution of the equation with R=¢(x) on the right side can at once be verified to be Ayi+BY2+Yiuy2v, where u, v respectively denote the integrals
u = f y2sb(x) (Yl'Y2  y2tyi ldx, v = J y1O(x) (Yi'Y2 Y2')l)ldx. The equation
d2 d
d—x2+Pdx+Qy =0,
by writing y =v exp. (If Pdx), is at once seen to be reduced to 2 I dv
dx2+Iv=o, where I=Qa j 4P2. If = v dx' the equation d2v do
dx2+Iv ...so becomes (5 = I +n2, a nonlinear equation of the first order.
More generally the equation
dx =A+Bn+Cn2'
where A, B, C are functions of x, is, by the substitution
r dy
n=Cy dx'
reduced to the linear equation
ax  (+c c&dC\
') d.1'ACy =o.
The equation
chi =A+Bn+Cn2,
known as Riccati's equation, is transformed into an equation of the same form by a substitution of the form n= (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 .no, the
substitution n=no1/Y enables us at once to obtain the general solution; for instance, when
2B =ax log (C) '
a particular solution is no= d (—A/C). This is a case of the remark, often useful in practice, that the linear equation
d2 ldcts dy
0(4712 +pax +µY°,
whereµ is a constant, is reducible to a standard form by taking a new independent variable z= f dx[cts(x)]i.
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
dt =ax+by+c, dt =a'x+b'y+c',
wherein the coefficients a, b, c, a', b', c', are constants. To integrate these, form with the constant X the differential coefficient of z=x+Xy, that is dz/dt=(a+Xa')x+(b+Xb')y+c+ac', the quantity a being so chosen that b+Xb'=X(a+Xa ), so that we have dz/dt = (a+Xa')z+c+ac'; this last equation is at once integrable in the form z(a+Xa')+c+Xc'=Ae(a+aa')e where A is an arbitrary constant. In general, the condition b+ab'=a(a+Xa') is satisfied by two different values of X, say X2; the solutions corresponding to these give the values of x+aly and x+A2y, 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 a are equal, that is, if (ab')2+4a'b=o, the method described gives only one equation, expressing x+ay 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+ra')a 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
a=aat+;
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 d2x/dt2 = 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
a (µY) =ay(AZ), &c.,
and hence
X(ay az)+Z'(ez~ ax) +Z (ax _aX)
conversely it can be proved that this is sufficient in order that µ may exist to render µ(Xdx+Ydy+Zdz) a perfect differential; in particular it may be satisfied in virtue of the three equations such as
aZ aY ay az'
•'n, which case we may take µ=i. Assuming the condition in
its general form, take in the given differential equation a plane section of the surface 4)=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=o; let 4'(x, y, z =constant, be its integral, the constant z entering, as a rule, in ¢ because it enters in X and Y. Now differentiate the relation ¢(x, y, z) =f(z), where/ is a function to be determined, so obtaining
azdx+aydy+ \az—d~) dz=o;
there exists a function a of x, y, z such that ax=aX, ay=aY,
because +' =constant, is the integral of Xdx+Ydy=o; we des+re to prove that f can be chosen so that also, in virtue of 4'(x, y, z) =fAz), we have
az d =QZ, namely d =az QZ;
Pax+Qay = R,
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
au au au av av av Pax}Qayf+n=o and Pz}Qay{Raz=o.
Then if F(x, y, z) =o be a relation satisfying the original differential equations, this relation giving rise to
OF aF az OF aF az aF aF aF
ax +a ax =o and ayj+az ay =o, we have P Ox +Qay +Raz =o.
It follows that the determinant of three rows and columns vanishes whose first row consists of the three quantities aF/ax, aF/ay, aF/az, whose second row consists of the three quantities au/Ox, au/ay, au/0z, whose third row consists similarly of the partial derivatives of v. The vanishing of this socalled 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 ¢(u, v) =o can easily be proved, in virtue of the equations satisfied by u and v, to lead to
Pdx+Qdy = R.
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,
z
ate =u+vx+wdt'
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 = o,
where X, Y, Z are functions of x, y, z. We consider only the case in which there exists an equation 0(x, y, z) =C whose differential ax+dy+—a——dz=o is equivalent with the given differential equation; that is,µ being a proper function of x, y, z, we assume that there exist equations
a =µX a =µY a=µZ. Ox 'ay 'az '
these equations require
if this can be proved the relation 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 ¢q(x, y, z) =f(z), the function
—uZ can be expressed in terms of z only. Now in consequence ax
of the originally assumed relations, ax_µX'ay= Y'az—µZ'
we have
ax ax µ ay ay'
and hence 00 a (34700
00_4 ads_
ax ayay ax
this shows that, as functions of x and y, 4' is a function of cis (see the note at the end of part i. of this article, on Jacobian determinants), so that we may write 4'=F(z, 4)), from which
aF a,' OF aFa4aF a aF a>G OF =a~;thenaz az+amaz _ az+µI&Z=az+aZoraz—aZ=dz;
in virtue of ¢(x, y, z) =f(z), and 4'=F(z, 4,), the function 4, can be written in terms of z only, thus aF/az 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
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
0_2_~ +
ax apax and ay _apay ay'
_a1ao=tLQ.
assay ayax axay'
by hypothesis ap/ax is not identically zero; therefore if the Jacobian determinant of p and q in regard to x and y is zero identically, so is aQ/ay, 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 aF/ax, aF/ay, aF/az in the determinant are all identically zero, namely the three determinants such as
auav auav
ay azaz ay'
then by the case of two variables considered above there exist three functional relations ,1(u, v, x) = o, +,2(u, v, y) = o, v, z) = o, of which the first, for example, follows from the vanishing of
auav auav
a'  oz Oz ay
We cannot assume that xis absent from Wi, or y from 11,2, or z from ,ka; but conversely we cannot simultaneously have x entering in W1, and y in ¢z, and'I' z in Os, or else by elimination of u and v from the three equations 41=0, 4' =o, ¢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 aF/ax, aF/ay, aF/az 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
auav auav
ay  az az ay
is not zero. Then from the equations u(x, y, z) =u, v(x,y,z) =vwecan 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
auav _auav
aya az ay
was zero), and so write F in the form F(x, y, z) ='F(u, v, x). We then have
aF a'Fau ads 0v a'F OF aslsau a'Fav aF aiDau a'Fav
exx auax+av ax+ax' ay —auay+av ay' az auaz+av az' thereby the Jacobian determinant of F, u, v is reduced to
a' auavauavl
ax ayaz az ay/ '
by hypothesis the second factor of this does not vanish identically; hence a'F/ax=o identically, and'F 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 Transformationtheories, Functiontheories; 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 transformstiongroups 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 existencetheorems 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 functiontheory 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 functiontheory 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 x°, 3707 .. of values of the independent variables x, y, ... of the function, at whatever point of the range of values for x, y, ... under consideration x°, 310, .. may be chosen, the function can be expressed as a series of positive integral powers of the differences x—x°, 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 x°, y°, , many results provisionally stated in the transformation theories would be unmeaning or incorrect. If, then, we have a set of k functions; f 1 . . .fk of as independent variables x, . . . xa, 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 rth row has the constituents df /dx,, ...df,Jdx,,; the justification being in the theorem, which we assume, that if the determinant involving, for instance, the first k columns be not zero for xi=xi ... xn=xn°, and the functions be developable about this point, then from the equations f1=ci, ...fk=ck we can express x1, ... xk by convergent power series in the
differences xk+1—xk +1, • • • xn—xn°, and so regard xi, . . xk as functions of the remaining variables. This we often express by
saying that the equations f f = c1, . . . fk= ck can be solved for x1, ... x,,. The explanation is given as a type of explanation
often understood in what follows.
We may conveniently begin by stating the theorem: If each of the n functions 4,i, ... 4,a of the (n +1) variables x,, . . . xat be developable about the values x,°, . . . x„°t°, the n differential equations of the form dx;/dt=4;(tx1, . . . x,,) are satisfied Ordinary
by convergent power series equations
x, =x,°+(t —t°)A,i+(t—tu)2Ar2+ . of the first
reducing respectively to x1°, ...x,.° when t=t°; and the order. only functions satisfying the equations and reducing respectively to x1°, .. X,,° when t =t°, are those determined by continuation of these series. If the result of solving these n equations for x,°, . . . xn°be written in the form w1 (x,, ... xn.t) = x1°, ...wa (xi, ... xat) = x,,°O, single it is at once evident that the differential equation
df/dt+¢1df/dxi+. . .+4'.df/dxn=o' homogene
possesses n integrals, namely, the functions w1, . wa, ous partial
which are developable about the values (xi .... x„°t) and equation reduce respectively to x 1, . x„ when t = P. And in fact it of the first has no other integrals so reducing. Thus this equation order. also possesses a unique integral reducing when t=t° to an arbitrary function >G(xi, . . . x~, this integral being 4'(w1, . . . wa.). Conversely the existence of these principal integrals w1, . . . to,. of the partial equation establishes the existence of the specified solutions of the ordinary equations dx;/dt=4t. 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°, &c., and consider the equation a(xyt)df/dx+b(xyt)df/dy=df/dt, wherein the functions a, b are developable about x=o, y=o, t=o; say
a(xyt) =a°Hai +t2a2/2!+..., b(xyt) =b°+tbi+t'b2/2!+..., so that
ad/dx+bd/dy=S°+t, i+I1'eh+ ... ,
where S =a,d/dxf.b,d/dy. In order that
.f=p°+tpi+t21p2/2!+ . .
these give
wherein p., pi ... are power series in x, y, should satisfy the equation, it is necessary, as we find by equating like terms, that
Pi =b°p°,P2=&Pi+Sip°, &c. Proof and in general
of the
existence where sr=(s!)/(r!) (s—r)!
of 'ate
Now compare with the given equation another equation grabs' 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=P°+tPi+t2P2/2!+ ... ,
wherein the coefficients in P. are real and positive, and each not less than the absolute value of the corresponding coefficient in p,; then putting A,=Ard/dx+Brd/dy we obtain necessary equations of the same form as before,. namely,
Pi=O°P°, P2=0°P1+A1P .
and in general P,+1 ..+z .P°. These give for every coefficient in P,+i an integral aggregate with real positive coefficients of the coefficients in P„ P,_l, . . . , Ps 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 in terms of the coefficients in p., . '. Ps and the coefficients in a and b. Hence as the coefficients in P. and also in A, B are real and positive, it follows that the values obtained in succession for the coefficients in Pi, 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.+i is, in absolute value, not greater than the corresponding coefficient in P,+i. Thus if the series for F be convergent, the series for f will also be; and we are thus reduced to (I), 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 P°+tPi+t2P2/2!+ . . in which the coefficients in P° are real and positive, and each not less than the absolute value of the corresponding coefficient in N. 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
1
takeA=B=M (1—1 yY) (I— ) —i and obtain
F = r — (r —x —y) CI 4M R (
y y) log (i — R)
r ys}1= 60p,+s1SlPa—1 +S282P,_2+ . +S,y°,
and that this solves the problem when x, y, t are sufficiently small for the two cases p° = x, p° = 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 then equations d y/dx = yi, d yi /dx =y2 .. ,
dYn—1 /dx = P—Piyn—i— ...—PnY
but in fact any simultaneous system of ordinary equations is reducible to a system of the form
dx;/dt = 4 (txi, . x.).
Suppose we have k homogeneous linear partial equations of the first order in n independent variables, the general equation being S i m u t t a a e  a.idf/dxi+...+ao,.df/dxn.=o, where 1 ,.. ... k, and that
linear we desire to know whether the equations have common ous
partial
solutions, and if so, how many. It is to be understood loan. that the equations are linearly independent, which implies
equat
that k 7n and not every determinant of k rows and columns is identically zero in the matrix in which the ith element of the oth row is aai(i = i, n, a = i, . . . k). Denoting the left side of the 0th equation by Paf, it is clear that every common solution of the two equations Paf =o, Ppf =o is also a solution of the equation Pp(Pof)—Pu(Ppf)=o. We immediately find, however, that this is also a linear equation, namely, EHidf/dx; =owhere H, =Ppaa, Paapi, 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 is 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 socalled complete system is is, the equations give df/dxi =o . . . df/dxn=o, leading .to the nugatory result f =a constant. Suppose, then, the number of this system to be r,
End of Article: DIFFERENTIAL EQUATION 

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