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DIFFERENTIAL EQUATION

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Originally appearing in Volume V08, Page 234 of the 1911 Encyclopedia Britannica.
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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 there-fore 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)dy-ff(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 . .)uo J } Xmi-1 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=cx-bf(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 co-ordinates 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 ex-pressing 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 the-other 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(0-B)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 Re-9''dxdx, or, say, y= (A-f Bx)eox-1-U, 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 (A-Ao)ex+(B-BOO' +u, or {A-As+(B-Bo)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" (x-f-PY) =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+ ... +Arxr-1)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 D-l(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 D-2(e4zu), which means D-1fD-1(e°xu)], is equal to D-1(e°xz) and hence to e°x(D+a)-'z, which we write e°x(D+a)-2u; proceeding thus we obtain D-a(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, ~(trf-a ' by first writing it as a sum of partial fractions, or otherwise, could be identically written in the form Kit-'+ Ka-it' +...+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-''+ ... +KiD-i+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-''+ ... +KiD-i+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 (D2-l-I)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)e-cx 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=xD-2(I+iD-1.D2-aiD3+ D4+AiD3...)x3, or -4etx(2isx5+iix4-4x3-2ix2+Vx+ii); the real part of this is .220 cos x+4(4x4-axe+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+B-9 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„01-1 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 da-2y dy xa I I'txa-ldxa i+... +Pn-xxdx+Pay, where Pi, . . . P. are constants, can be reduced to the case considered above. Writing x=et we have the identity dmu xmclxf°=a(9-I)(0-2). ..(9-m+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+Pdx-I -QY = o, there being no relation of the form myl-Eny2=k, where m, n, k are constants, it is easy to see that d dx(Yi'Y2-y1Y2) = P(Yi'Y2-Y1Y2'), so that we have yl'Y2-Yiy2' =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+Yiu-y2v, 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=Q-a j- -4P2. If = -v dx' the equation d2v do dx2+Iv ...so becomes (-5 = I +n2, a non-linear 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=no-1/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 (a-b')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-}-Qay-f-+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 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 ¢(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 ay-ay 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 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 transformstion-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 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 r-th 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 develop-able 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/dx-f.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 i-th element of the o-th row is aai(i = i, n, a = i, . . . k). Denoting the left side of the 0-th 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 so-called 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, d,p do dpi dxi+pi dz - dpi (dxi+pi dz) It can at once be verified that for any three functions[f[4IG]]+[¢[,pf]] +[4'[f4.]]=dz[4.4']+d [¢f]+c- [f4,], which when f, 4,, 4, do not contains becomes the identity (f (¢,p)) + (4.(,pf)) + (p( pi)) = o. Then, if XI,. ..Xs, P1, P. be such functions of xi, . . . x,,, p1 .. pn that P1dX1 +. . . +PndXn is identically equal to pidx1+... +pndxn, it can be shown by elementary algebra, after equating coefficients of independent differentials, (I) that the functions X1, ... Pn are independent functions of the 2n variables x1, . . . P,,, so that the equations x'i =Xi, p'i = Pi'can be solved for . . . x,., pl, . . . pn, and represent therefore a transformation, which we call a homogeneous contact transformation; (2) that the Xi, ... Xn are homogeneous functions of Pi,... Ps of zero dimensions the Pr, ... P. are homogeneous functions of pi, .:. pn of dimension one, and the zn(n-r) relations (XiX;) =o are verified. So also are the n2 relations (PiXi) =1, (PiXi) =o, (PiPi) =0. Conversely, if X1, ... Xn be independent functions, each homogeneous of zero dimension in pl, ... pn satisfying the In(n—t) relations (XiX;) =0, then P1, ... Pn can be uniquely determined, by solving linear algebraic equations, such that P1dX1+...+PndXn =pidx1+. +pndxn. If now we put n+i for it, put z for x,, i, Z for Xn}1, Qi for—Pi./P,,+i, for i=s, ... n, put qi for—p;/pn}1 and v for qn+i/Qn}i, and then finally write PI,. . . Pn> Po" . Ps for Q1, ... Qn, q1, ... qn, we obtain the following results: If ZX1 ... Xs, P1, ... Ps be functions of z, xi,. . . xn, Pi, ... p,,, such that the expression dZ-P1dX1-...-PndXn is identically equal to c(dz-pidxi-... pndxn), and Q not zero, then (I) the functions Z, X1, . . . Xs, Pi, ... Ps are independent functions of z, x1, . . xn, (PI, ... is., so that the equations z' = Z, x'i =Xi, p'i = Pi can be solvefor z, x1, ... x,., p1>. - -Ps and determine a transformation which we call a (non-homogeneous) contact transformation; (2) the Z, X1, ... L, verify me fn(n+I) identities [ZX,]=o, [X,Xi]=o. And the further identities [P,X,]=a, [P,Xi1=o, [P,Z] =aPi, [P,Pi]=o, gal= adZ — o2, [Xi a] = adX, [ P u r l = cr- P, dz dz ' dz are also verified. Conversely, if Z, Xi, ... X. be independent functions satisfying the identities [ZXi] o, [X,X,] =o, then e, other than zero, and P1, ... P. can be uniquely determined, by solution of algebraic equations, such that dZ—P1dX1— ...—P„dX„=e(dz-pidxi— ... Finally, there is a particular case of great importance arising when =1, which gives the results: (I) If U, XI, ... X,,, PI, . . . P. be 2n+I functions of the 2n independent variables x1, . . . . . . satisfying the identity dU+Pidxi+ . . . =pidxi+ . +pndx,., then the 2n functions P1, . . . P,., X,, . . . X„ are independent, and we have (X,Xi)=o,(X:U)=&X,,(P,X,)= I,(P:Xi)=o, (P,Pi) o,=(PiU) -{-P, =SP,, where S denotes the operator pid/dp,+ . . . +p„d/dp,,; (2) If X,, . . . X„ be independent functions of x1, . . x,,, p,, . . . p,,, such that (X,Xi) =o, then U can be found by a quadrature, such that (X,U)=SX,; and when X,, ... X,,, U satisfy these 2n(n+I) conditions, then F1, ... P„ can be found, by solution of linear algebraic equations, to render true the identity dU+P,dX, I ... FP dX =y,dxi+...+p„dx,,; (3) Functions XI, . . . . . . P„ can be found to satisfy this differential identity when U is an arbitrary given function of p1, .. . p,,; but this requires integrations. In order to see what integrations, it is only necessary to verify the statement that if U be an arbitrary given function of x1, ... xn, pl, . . . p,,, and, for r G(x, y) will necessarily be associated with the equations p=dz/dx, q=dz/dy, and z will satisfy the equation in its original form. We have previously seen (under Pfaffian Expressions) that if five variables x, y, z, p, q, otherwise independent, be subject to dz—pdx—qdy = o, they must in fact be subject to at least three mutual relations. If we associate with a point (x, y, z) the plane Z—z=p(X—x)+q(Y—y) passing through it, where X, Y, Z are current co-ordinates, and call this association a surface-element; and if two consecutive elements of which the point(x+dx, y+dy, z+dz)of one lies on the plane of the other, for which, that is, the condition dz= pdx+qdy is satisfied, be said to be connected, and an infinity of connected elements following one another continuously be called a connectivity, then our statement is that a connectivity consists of not more than oo 2 elements, the whole number of elements (x, y, z, p, q) that are possible being called oo 6. The solution of an equation F(x, y, z, dz/dx, dz/dy) =o is then to be understood to mean finding in all possible ways, from the o0 4 elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) =o a set of o0 2 elements forming a connectivity; or, more analytically, finding in all possible ways two relations G = o, H = o connecting x, y, z, p, q and independent of F = o, so that the three relations together may involve dz = pdx+qdy. Such a set of three relations may, for example, be of the form z— &(x, y), p=d¢/dx, q=d¢/dy; but it may also, as another case, involve two relations z= t/i(y), x= %',(y) connecting x, y, z, the third relation being (y) = p¢'1(y)+q, the connectivity consisting in that case, geometrically, of a curve in space taken with oo 1 of its tangent planes; or, finally, a connectivity is constituted by a fixed point and all the planes passing through that point. This generalized view of the meaning of a solution of F = o is of advantage, moreover, in view of anomalies otherwise arising from special forms of the equation itself. For instance, we may include the case, some- Meaning times arising when the equation to be solved is obtained of a soap by transformation from another equation, in which F tion of the does not contain either p or q. Then the equation has equation. o0 2 solutions, each consisting of an arbitrary point of the surface F= o and all the o0 2 planes passing through this point; it also has o0 2 solutions, each consisting of a curve drawn on the surface F = o and all the tangent planes of this curve, the whole consisting of o0 2 elements; finally, it has also an isolated (or singular) solution consisting of the points of the surface, each associated with the tangent plane of the surface thereat, also 00 2 elements in all. Or again, a linear equation F = Pp+ Qq— R= o, wherein P, Q, R are functions of x, y, z only, has o0 2 solutions, each consisting of one of the curves defined by dx/P = dy/Q = dz/R taken with all the tangent planes of this curve; and the same equation has 00 2 solutions, each consisting of the points of a surface containing 0o ' of these curves and the tangent planes of this surface. And for the case of n variables there is similarly the possibility of n+x kinds of solution of an equation F(xi, ... z, Pi, ... I%) =o; these can, however, by a simple contact transformation be reduced to one kind, in which there is only one relation z'= >'(x',, . . . x'„) connecting the new variables x'1, . . . x',,, z' (see under Pfaffian Expressions); just as in the case of the solution z='(y), x='',(y), 1//(y)=p'P'i(y)+q of the equation Pp+Qq=R the transformation z'=z—px, x'=p, p'= —x, y'=y, q'=q gives the solution z'='//(y')+x'1/'1(y'), p'=dz'/dx', q' = dz'/dy' of the transformed equation. These explanations take no account of the possibility of p and q being infinite; this can be dealt with by writing p= —u/w, q= —v/w, and considering homogeneous equations in u, v, w, with udx+vdy+wdz=o as the differential relation necessary for a connectivity; in practice we use the ideas associated with such a procedure more often without the appropriate notation. In utilizing these general notions we shall first consider the theory of characteristic chains, initiated by Cauchy, which shows well the nature of the relations implied by the given differential equation; the alternative ways of carrying out the necessary integrations are suggested by con- sidering Order of the method of Jacobi and Mayer, while a good the ideas. summary is obtained by the formulation in terms of a Pfaffian expression. Consider a solution of F =o expressed by the three independent equations F =o, G =o, H =o. If it be a solution in which there is more than one relation connectingx,y,z, let new variables x' ,y' ,z' ,p' ,q' be introduced, as before explained under Pfaffian Ex- Charao• pressions, in which z' is of the form terlstk z'=z—p1x1—• .. —p,x,(s=I or 2), chains. so that the solution becomes of a form z'=y'(x'y'), p' =dOdx', q' =d¢ldy', which then will identically satisfy the trans-formed equations F' = o, G' = o, H' = o. The equation F' = o, if x',y,'z' be regarded as fixed, states that the plane Z- z=,p'(X-x')+q'(Y-y') is tangent to a certain cone whose vertex is (x', y', z'), the consecutive point (x'+dx', y'+dz', z'+dz') of the generator of contact being such that dx''dp,=dy'(dg =dz''(p'dp,-+-q'dq,)' Passing in this direction on the surface z'=,/,(x', y') the tangent plane of the surface at this consecutive point is (p'+dp', q'+dq'), where, since F'(x', y', d^Pldx', d¢ldy) =o is identical, we have dx' (dF'/dx' +p'dF'/dz') +dp'dF'/dp' = o. Thus the equations, which we shall call the characteristic equations, dF' dFdF' +q`dF' , _dF' ,dF' dx'~dp'=dy / dq,dz (pap,dq') =dp I ( dx'—p az') dF' ,dF') dq'/ (—'—q dz are satisfied along a connectivity of col elements consisting of a curve on z' =0(x', y') and the tangent planes of the surface along this curve. The equation F'=o, when p', q' are fixed, represents a curve in the plane Z-e'=p'(X-x')+q'(Y-y') passing through (x', y', z'); if y' =by', z'+Sz') be a consecutive point of this curve, we find at once Sx' ( dF' dF' +p' ) +ay'(Try dF' +q',) =o; thus the equations above give Sx'dp'+Sy'd '=o, or the tangent line of the plane curve, is, on the surface z' =+'(qx', y'), in a direction conjugate to that of the generator of the cone. Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x'o, y'o, z'o, p',, q'o, about which all the quantities F', dF'/dp', &c., occurring in the denominators, are developable, to define, from the differential equation F' =o alone, a connectivity of col elements, which we call a characteristic chain; and it is remarkable that when we transform again to the original variables (x, y, z, p, q), the form of the differential equations for the chain is unaltered, so that they can be written down at once from the equation F =o. Thus we have proved that the characteristic chain starting from any ordinary element of any integral of this equation F=o consists only of elements belonging to this integral. For instance, if the equation do not contain p, q, the characteristic chain, starting from an arbitrary plane through an arbitrary point of the surface F =o, consists of a pencil of planes whose axis is a tangent line of the surface F =o. Or if F =o be of the form Pp+Qq=R, the chain consists of a curve satisfying dx/P=dy/Q=ds/R and a single infinity of tangent planes of this curve, determined by the tangent plane chosen at the initial point. In all cases there are 00 2 characteristic chains, whose aggregate may therefore be expected to exhaust the o0 4 elements satisfying F =o. Consider, in fact, a single infinity of connected elements each satisfying F =o, say a chain connectivity T, consisting of 'elements specified by xo, yo, zo, po, qo, which we suppose expressed as Complete functions of a parameter u, so that lntearal U. = dzo/du — podxo/du —godyo/du con- is everywhere zero on this chain; further, suppose that strutted each of F, dF/dp, . dF/dx+pdF/dz is developable with about each element of this chain T, and that T is not a character, characteristic chain. Then consider the aggregate of the characteristic chains issuing from all the elements of T. chains. The o0 2 elements, consisting of the aggregate of these characteristic chains, satisfy F =o, provided the chain connectivity T consists of elements satisfying F=o; for each characteristic chain satisfies dF=o. It can be shown that these chains are connected; in other words, that if x, y, z, p, q, be any element of one of these characteristic chains, not only is dz/dt — pdx/dt —qd/dt=o, as we know, but also U=dz/du—pdx/du—gdy/du is also zero. For dU d _ d dz dx dy ~_Tl (dz ~_ dx —qdu) du WI rdt qdt dppdxp dx dy —q dy =du dt —dt N-1-du dt dt du' which is equal to d dF dx (dF dF ddq~ dF dF dF dF u~+~ (~+per) +duT +du (dy+ dz) =- d.U. Asa- is a developable function of t, this, giving t dF U=Uo exp (_ (to dt), shows that U is everywhere zero. Thus integrals of F=o are obtainable by considering the aggregate of characteristic chains issuing from arbitrary chain connectivities T satisfying F=o; and such connectivities T are, it is seen at once, determinable without integration. Conversely, as such a chain connectivity T can be taken out from the elements of any given integral all possible integrals are obtainable in this way. For instance, an arbitrary curve in space, given by xo=o(u), yo=4(u),zo=,G(u), determines by the two equations F(xo, yo, so, p,, qo,) =o, i'(u) = poo'(u)+qo¢'(u) such a chain connectivity T, through which there passes a perfectly definite integral of the equation F=o. By taking co 2 initial chain connectivities T, as for instance by taking the curves xo=o, yo=O, zo=~G to be the co 2 curves upon an arbitrary surface, we thus obtain oo 2 integrals, and so o0 4 elements satisfying F = o. In general, if functions G, H, independent of F, be obtained, such that the equations F = o, G = b, H =c represent an integral for all values of the constants b, c, these equations are said to constitute a complete integral. Then o0 4 elements satisfying F =o are known, and in fact every other form of integral can be obtained without further integrations. In the foregoing discussion of the differential equations of a characteristic chain, the denominators dF/dp, ... may be supposed to be modified in form by means of F =o in any way conducive to a simple integration. In the immediately following explanation of ideas, however, we consider indifferently all equations F=constant ; when a function of x, y, z, p, q is said to be zero, it is meant that this is so identically, not in virtue of F=o; in other words, we consider the integration of F =a, where a is an arbitrary constant. In the theory of linear partial equations we have seen that the integration of the equations of the characteristic chains, from which, operations as has just been seen, that of the equation F =a followsnecessary at once, would be involved in completely integrating for the single linear homogeneous partial differential equation int of the first order [Ff] =o where the notation is that Hoelo ' explained above under Contact Transformations. One F=a. obvious integral isf=F. Putting F =a, where a is arbi- trary, and eliminating one of the independent variables, we can reduce this equation [Ff] =o to one in four variables; and soon. Calling, then, the determination of a single integral of a single homogeneous partial differential equation of the first order in n independent variables, an operation of order n-r, the characteristic chains, and therefore the most general integral of F=a, can be obtained by successive operations of orders 3, 2, I. If, however, an integral of F=a be represented by F = a, G = b, H =c, where b and c are arbitrary constants, the expression of the fact that a characteristic chain of F=a satisfies dG=o, gives [FG]=o; similarly, [FH]=o and [GH]=o, these three relations being identically true. Conversely, suppose that an integral G, independent of F, has been obtained of the equation [Ff] =o, which is an operation of order three. Then it follows from the identity [f[q5+y]]+[4[tPfI]+[WO]]=a[k]+az[*f]+7-4 .41 before remarked, by putting 0=F, =G, and then (Ff]=A(f),[Gf]=B(f), that AB (f)-BA(f) = ddzB (f)-sA(f ), so that the two linear equations [Ff]=o, [Gf] =o form a complete system; as two integrals F, G are known, they have a common integral H, independent of F, G, deter-minable by an operation of order one only. The three functions F, G, H thus identically satisfy the relations (FG] = [GH] = [FH] =o. The o0 2 elements satisfying F =a, G =b, H =c, wherein a, b, c are assigned constants, can then be seen to constitute an integral of F = a. For the conditions that a characteristic chain of G =b issuing from an element satisfying F=a, G=b, H=c should consist only of elemehts satisfying these three equations are simply[FG] =o,[GH] = o. Thus, starting from an arbitrary element of (F=a, G=b, H=c),we can single out a connectivity of elements of (F=a, G=b, H=c) forming a characteristic chain of G=b; then the aggregate of the characteristic chains of F =a issuing from the elements of this characteristic chain of G =b will be a connectivity consisting only of elements of (F=a, G=b, H=c), and will therefore constitute an integral of F=a; further, it will include all elements of (F =a, G =b, H =c). This result follows also from a theorem given under Contact Transformations, which shows, moreover, that though the characteristic chains of F=a are not determined by the three equations F =a, G = b, H = c, no further integration is now necessary to find them. By this theorem, since identically [FG] = [GM= [FH] =o, we can find, by the solution of linear algebraic equations only, a non-vanishing function u and two functions A, C, such that dG —AdF —CdH = v(dz — pdz — qdy) ; thus all the elements satisfying F = a,G = b,H = c, satisfy dz = pdx+qdy and constitute a connectivity, which is therefore an integral of F=a. While, further, from the associated theorems, F, G, H, A, C are independent functions and [FC]=o. Thus C may be taken to be the remaining integral independent of G, H, of the equation [Ff]=o, whereby the characteristic chains are entirely determined. When we consider the particular equation F =o, neglecting the case when neither p nor q enters, and supposing p to enter, we may express p from F = o in terms of x, y, z, q, and then eliminate it from all other equations. Then instead of the equation [Ff]=o, we have, if F=o give p=0(x, y, z, q), the equation =- ( +tbdz) +(+q) — (dy+q dZ) = o, moreover obtainable by omitting the term in df/dp in [p-v', f] =o. Let xo, yo, zo, qo, be values about which the coefficients in The sinee this equation are developable, and let n, w be the equation principal solutions reducing respectively to z, y and q F-o and when x = xo. Then the equations p = = so, n =yo, co = qo mama represent a characteristic chain issuing from the element formulaxo, yo, zo, 1 o, go; we have seen that the aggregate of Hors. such chains issuing from the elements of an arbitrary chain satisfying dzo — podxo —MN = o constitute an integral of the equation p = '. Let this arbitrary we have chain be taken. so that x° is constant; then the condition for initial values is only dz°—q°dy° =0, and the elements of the integral constituted by the characteristic chains issuing therefrom satisfy d3 —wdn=o. Hence this equation involves dz—pdx—qdy=o, or we have dz—pdx-qdy=a(di--wdn), where o is not zero. Conversely, the integration of p= p is, essentially, the problem of writing the expression dz—pdx—qdy in the form o(di'—wdn), as must be possible (from what was said under Pfaffian .. xpressions). To integrate a system of simultaneous equations of the first order Xi =al, . . . X,. = ar in n independent variables xi, . . xn and and one dependent variable z, we write pi for dz/dxi, &c., Systesystemns. and attempt to find n+i —r further functions Z, Xr+i e eq the first .. . Xsuch that the equations Z = a, X, =a, (i = i, . . . n) order. involve dz—pidxi—...-pndxn=o. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation X; =a„ thus each of the expressions [X;X;] must vanish in virtue of the equations expressing the integral, and we may without loss of generality assume that each of the corresponding 1r(r— I) expressions formed from the r given differential equations vanishes in virtue of these equations. The determination of the remaining n+I —r functions may, as before, be made to depend on characteristic chains, which in this case, however, are manifolds of r dimensions obtained by integrating the equations [Xif]=o, . . . [Xrfl=o; or having obtained one integral of this system other than XI, Xr, say Xr+i, we may consider the system [Xif] =o, . . . [Xr+If]=o, for which, again, we have a choice ; and at any stage we may use Mayer's method and reduce the simultaneous linear equations to one equation involving parameters; while if at any stage of the process we find some but not all of the integrals of the simultaneous system, they can be used to simplify the remaining work; this can only be clearly explained in connexion with the theory of so-called function groups for which we have no space. One result arising is that the simul- taneous system pI=¢I, . pr=Or, wherein . Ps are not involved in . Or, if it satisfies the ir(r - i) relations [p;—p;, pi—4';]=o, has a solution z=p(xi, . xn) pi=dp/dxi, . . . yyn=dp/dxn, reducing to an arbitrary function of x,+i, . . . xn only, when xi=xi°, . . . x,= x,° under certain conditions as to developability; a generalization of the theorem for linear equations. The problem of integration of this system is, as before, to put
End of Article: DIFFERENTIAL EQUATION
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