See also:

• DIFFERENTIAL

(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 See also:

• FAMILY

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 See also:

• MAIN (from the Aryan root which appears in " may " and " might," and Lat. magnus, great)
• MAIN (Lat. Moenus)

+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, See also:

• A2(z)

To See also:

• COMPLETE

. +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 See also:

• TERM

(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 (See also:

• EAR (common Teut.; O.E. are, Ger. Ohr, Du. oor, akin to Lat. auris, Gr. ovs)

(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 See also:

• RICCATI, JACOPO FRANCESCO, COUNT (1676-1754)

In general, the See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

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 See also:

• NOTE
• NOTE (Lat. nota, mark, sign, from noscere, to know)

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 See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

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 See also:

• FACTOR (from Lat. facere, to make or do)

. .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 See also:

• MATRIX

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 See also:

• PRINCIPAL

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 See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

Hence as the coefficients in P. and also in A, B are real and positive, it follows that the values obtained in See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

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 See also:

• LIMITATION, STATUTES OF

It is convenient, then, from the complete system of r equations to form r new equations by solving separatelyfordf/dxi, , df/dxr; suppose the general equation of the new system to be Qoif=df/dxx+ca,r+tdf/dx,+i+ . .. +cdndf/dxn=o(o=I, . r). Then it is easily obvious that the equation QpQaf-QaQpf =o contains only the differential coefficients off in regard to xr+1 ... x,, as it is at most a linear function of Qif, . . . Qrf, it must be identically zero. So reduced the, system is called a Jacobian system. Of this system Q1f =o has n—r principal solutions reducing re spectively.to x2, .. x,, when xi=xi Jacobean systems. , and its form shows that of these the first r-I are exactly x2 . . . xr. Let these n—r functions together with xi be introduced as is new independent variables in all the r equations. Since the first equation is satisfied by n—r of the new independent variables, it will contain no differential coefficients in regard to them, and will reduce therefore simply to df/dxi=o, expressing that any common solution of the s equations is a function only of the n-i remaining variables.

Thereby the investigation of the common solutions is reduced to the same problem for r—r equations in n—r variables. Proceeding thus, we reach at length one equation in n—r+I variables, from which, by retracing the See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

. . cos be the principal solutions of the Jacobian system, Qof =o, reducing respectively to xr+i, .:. is,, when xi=xi°, . xr=xr°, and the equations wr+i=x,°+i, . . . wn=xn° be solved for x,+1, . xn to. give xj=¢;(xi, . . . xr, x,°+i, . x,,°),these values solve the total equations and reduce respectively to xr +i, ...x, when xi=xi . . . xr=xr . And the total equations have no other solutions with these initial values. Conversely, the existence of these solutions of the total equations can be deduced a priori and the theory of the Jacobian system based upon them. The theory of such total equations, in general, finds its natural place under the heading Pfafan Expressions, below. A See also:

• PRACTICAL

In the general case the equations dxi =clidx1+• .cr;dx, See also:

• MAYER, JOHANN TOBIAS (1723-1762)
• MAYER, JULIUS ROBERT (1814-1878)

Platfian It is natural to inquire what relations among x, y, z, if any, Express are implied by, or are consistent with, a differential relation sions adx+bdy+cdx=o, when a, b, c are unrestricted functions of x, y, z. This problem leads to the consideration of the so-called Pfaffian Expression adx+bdy+cdz. It can be shown (I) if each of the quantities db/dz-dc/dy, dc/dx-da/dz, da/dy-db/dz, which we shall denote respectively by u23, u31, u12, be identically zero, the expression is the differential of a function of x, y, z, equal to dt say; (2) that if the quantity au2s+bua1+cuia is identically zero, the expression is of the form udt, i.e. it can be made a perfect differential by multiplication by the factor I/u; (3) that in general the expression is of the form dt+uldtl. Consider the matrix of four rows and three columns, in which the elements of the first row are a, b, c, and the elements of the (r+1)-th row, for r=1, 2, 3, are the quantities u,1, ur2, ura, where See also:

• U11

In all cases of a single Pfaffian equation we are thus led to consider what is implied by a relation dt-uldti-. . .umdt,n=o, in which Slagle t, u1, . um, t1 ., tin are, except for this equation, "ear independent variables. This is to be satisfied in virtue of Pfaffian one or several relations connecting the variables; these must involve relations connecting t, 11, . . . tin only, and equation. in one of these at least t must actually enter. We can then suppose that in one actual system of relations in virtue of which the Pfaffian equation is satisfied, all the relations connecting t,ti .. . In; only are given by t=4'(ts+)...tin), tt=4,1(ta+1...t,n), . . . t. 4's(ta}1 . . . tin) ; so that the equation dp-u1d,p1- . . . —uad,pe ua+idt,+1—...-u,ndtm=0 is identically true in regard to u1, . . . um, to}1 .

. tm; equating to zero the coefficients of the differentials of these variables, we thus obtain m-s relations of the form d,p/dt; uld¢1/dt; . . . -uad4a/dti-u; =o; these m-s relations, with the previous s+1 relations, constitute a set of m+1 relations connecting the 2m+I variables in virtue of which the Pfaffian equation is satisfied independently of the form of the functions +p, 4'i s. There is clearly such a set for each of the values s=o, s= 1, s=m–I, s=m. And for any value of s there may exist relations additional to the specified m+i relations, provided they do not involve any relation connecting t, 11, . . . tin only, and are consistent with the m-s relations connecting u1, . . . um. It is now evident that, essentially, the integration of a Pfaffian equation a1dx1+ . . . +andxn=0, wherein a1, . . . an are functions of xi, xn, is effected by the processes necessary to bring it to its reduced form, involving only independent variables. And it is easy to see that if we suppose this reduction to be carried out in all possible ways, there is no need to distinguish the classes of integrals corresponding to the various values of s; for it can be verified without difficulty that by putting t'=t—ultl—.

.'tlatar t'1=nlr . . . t',=nar u'1=—t1, u'a= t'a+1=ts}1, . . =trn, u's}1=ua}1, . . . u'm=urn, the reduced equation becomes changed to dt'-u'ldt'1-. . u'rndt'rn=o, and the general relations changed to /t'=,p(t'.+1, . . . ern) . . . t'm)–. t'eY s(t's+1, ... t'm), _0, say, together with u'1= d4/dt'1r , u'n,. = dq,/dt'rn,which contain only one relation connecting the variables t', ... t',n only. This method for a single Pfaffian equation can, strictly speaking, be generalized to a simultaneous system of (n-r) Pfaffian equations dx;=clidx1+. . .+c,idx, only in the case already treated, See also:

• MINA, FRANCISCO ESPOZ Y (1781-1836)

. be identically zero, then Ou+co,r}1 r}1+• +cu,n4n =o, or 4, satisfies the r partial differential equations previously associated with the total equations; when these are not a complete system, but included in a complete system of r—µ equations, having therefore n—r—µ independent integrals, the total equations are satisfied over a manifold of r+µ dimensions (see E. v. See also:

• WEBER, CARL MARIA FRIEDRICH ERNEST VON (1786–1826)
• WEBER, WILHELM EDUARD (1804-1891)

. . x,,, p1 .. pn that P1dX1 +. . . +PndXn is identically equal to pidx1+... +pndxn, it can be shown by elementary See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

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 See also:

• GREAT

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 See also:

• QUADRATURE (from Lat. quadratura, a making square)

. 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 <n, X1, . . . Xr be independent functions of these variables, such that (XIU)=SXo, (XpXa) =o, for p, a=I . . . r, then the r+t homogeneous linear partial differential equations of the first order (Uf)+Sf=o, (Xpf)=o, form a complete system. It will be seen that the assumptions above made for the reduction of Pfaffian expressions follow from the results here enunciated for contact transformations. We pass on now to consider the solution of any partial differential equation of the first order; we attempt to explain Partial certain ideas relatively to a single equation with any differen- number of independent variables (in particular, an tad vista- ordinary equation of the first order with one indetion of the pendent variable) by speaking of a single equation with first two independent variables x, y, and one dependent order variable z. It will be seen that we are naturally led to consider systems of such simultaneous equations, which we consider below. The central See also:

• DISCOVERY

We See also:

• NOTICE

. . 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 See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

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 See also:

• PENCIL (Lat. penicillus, brush, literally little tail)

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 See also:

• ARGUMENT