01 ENVELOPES AMD EXTRA1J013S LOCI Of DIFFERENTIAL EQUATIONS 01 0HD1R OH AM) HIGHER DEGREE t
APPROVED:
Major Professor
l£^L Minor Processor .(2 Direct©* of the Department of Mathematics
Dean of the Graduate School ON ENVELOPES AID EXTRANEOUS LOCI OF DIFFERENTIAL EQUATIONS OF ORDER Oil AID HIGHER DEGREE
THESIS
Presented to the Graduate Council of the North Texas State College In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF ARTS
By
Hal Burns Lane, Jr., B. A. I I
Denton, Texas August, 1961 TABLE 01 CONTENTS
Chapter I, INTRODUCTION ...... 1 II, THE c-DISCRIMINANT LOCI. 6 Envelopes Conditions for Some Loci Node-locus Cusp-locus Factors of the c-Discriminant Envelope Node-locus Cusp-locus III, THE p-DISCRIMINANT LOCI, . . . 28 Loci of Contacts of Parallel Tangents Factors of the p-Discriminant BIBLIOGRAPHY 4$
ill CHAPTER I
INTRODUCTION
It is found in various books that the differential equations of order one may "be implicitly written as
P(x,ytp) » 0, where p « dy/dx, and have a solution of f(x,y,c) ® 0. The solution, referred to as the general solution, describes a family of curves that are called solution curves; whereby the coordinates and the direction at each point on a solution curve will satisfy the differ- ential equation. Since these curves are obtained by assigning arbitrary values to c, the question is raised as to whether or not there could exist a curve whose points v/ould satisfy the differential equation, and yet not be one of the curves obtained in the general solution. Upon investigation, such a curve is found and is called the envelope of the differential equation. In addition to the envelope, other loci of points having unique properties are found associated with the differential equation and its solution curves. However, these loci are not neces- sarily solutions of the differential equation and are referred to as extraneous loci. The purpose of this paper is to examine the proper- ties of th© envelope and the extraneous loci associated with the solution curves of ordinary differential equa- tions of the first order and degree greater than one. The differential equation to be considered is of the form P(x,y,p) * 0 where p « dy/dx and the function F(x,y,p) is a polynomial in p. The general solution of F(xty,p) « 0 is a family of curves expressed by the relation f(x,yfc) • 0 where c is the parameter that determines a unique solution curve for each arbitrary value assigned it.
The existence of the solution of F(x,yfp) » 0 is assumed in this paper. In addition to assuming the basic relations in differ- ential calculus, certain theorems will be stated without proof since they are needed as basic tools in the develop- ment of this paper. Also, for simplicity of writing, the notation F will be used to denote dF/dx. Theorem 1.1. Iiet P(x,y,z) £§. a continuous function of the independent variables x, y, and z with partial deriva- tives Px, Fyt an| Fg. for the systeia of values (x0,yQtz0), i§t ^(x0ly0,z0) - 0 and ffp(ac0»y0>zQ) f 0. Then mark off an interval < z sucli that (x0»y0) for every (x,y) £n R the equation F(x,y,z) «0ig satisfied b^ ,iust one value of z in the interval z^ <; z <; z^. This value of z, which is denoted by z « f(xty), is a continuous function of x,y and possess- §M continuous partial derivatives f , f , and z « f(x_,y^), a JF O O v ,%e derivatives of f are by the equations 1 i\x + F zf x « 0 —an-d Fy + F zf y « 0. Theorem 1,2, If in a function F(x,y,z) the variables are not independent of one another. but are sub.iect to the condition F * 0, then the linear parts of the increments (F„dx, F..dy, F„dz) of the variables are likewise not inde- pendent of one another, but are connected by the condition dF » 0, that is, by the linear equation f dx + i' dy + F„dz » G.2 x y v z Theorem 1.5. Let a, 6. , JR. C our ant, Differential and Integral Calculus. (New York, 1936), II, 117-118 2lbid., p. 118. 55 ^Frederic H. Miller. Partial Differential Equations, (New York, 1941), p. 50. this set of values. Sup-pose further that the Jacoblan for this set of values. Then there exists one and only one system of continuous functions x » f(a+h,b+k,c+l) « f(atb,c) + (hd * kd + 18 )f + »,, T£ lb 15c + i Clad + kd + ld_)nf + 1 (h£_ + kd + 13jn+1f(x,y,z) n! da db de Cn+1)T dx Ty dz where in the last term the values x » a + eh, y « b + ek, and z « c +• el, for 0 < e <1, are substituted after differ- entiation.^ Theorem 1,5, known as Taylor's Theorem, is frequently written in a different form. This second form will b[e given since it is also used in this paper. Corollary 1,1. Let f(x,y,z) and all of its first and second partial derivatives be differentiabl® near (a,b,c), 4Ibid.. p. 51. 5 C. A. Stewart, Advanced Calculus» (London, 1946). pp. 45-46. ~ f(x,y*z) - f(a,b,c) + fx(a,b,c)(x-a) + fy(a,b,c)(y-b) + a z -^2( »® ) (z~c) +• 1/2 fyyCx-j ,y-| tz1 )(x-a) + £yy(x^»y^» ^)(y—b )' + f x z 2 Z2^ i*yi» iKz-c) + 2fX3r(x19y1»z1)(x~a)(y-b) af^.y^Xx-aXz-c) + 2fy2(Xl,y1)Zl)(y-b)(a-c)] , gfeg..gg, (x-, fjj ,z-,) is some suitably chosen point between Jm «*!* wl* MumjAftiAi** (x,y»z) and. (a,b,c). CHAPTER II THE c-DISCHIMIHANT LOCI While this paper is primarily concerned with the fam- ily of curves associated with the general solution of a differential equation, the results of this chapter could he applied to any family of curves, f(x,y,c), whether the curves are solutions of a differential equation or not. The study of the envelope and extraneous loci of f(x,y,e) will utilize the c-discriminant and the idea of double points on a curve. It will be shown that the c-discriminant will reveal much about the various loci related to f(x,y,c). In fact it will be shown, as first 1 proposed by Cayley, that the c-discrlminant will, in gen- eral, contain the envelope-locus as a factor once and only once, the node-locus as a factor twice and only twice, and the cusp-locus as a factor thrice and only thrice. Before becoming involved with any discussion of the various loci of f(x,y,c), it will be necessary to define several terms. "^Cayley, **0n the Theory of the Singular Solutions of Differential Equations of the first Order,n Messenger of Mathematics, II (18?2), 11-12. Definition 2,1. $he c"discriminant is the relation by eliminating the c term between the two rela- tions t f(x,y,c) ® 0 and fc(x,y,c) - 0. Definition 2,2. A& envelope i& § whigh touches every member of the family, and which, at each point is touched by some member of the family« fwp curves are said. to touch if they have the same direction a£ a coiamon point. Definitlo n 2.3. A double point is a on a curve that satlsfle the conditions: f(x,y,c) « 0, fx(x,y»c) - 0, and f (x,y,c) - 0. fhe function f(x*y»c) is represented v M. $M. smm* Definition 2.4. A nodal point is a point on a curve that is a double point and has two directions of the curve given at the point. A continuous curve made up of such points is called the node-locus, Definition 2.5. A cuspidal point is a point on the curve that is a double point and has only one direction of the curve given at the point* 4 continuous curve made uj> of such points ig. called the cusp-locus. Definition 2.6. Tfeg tac-locus jj. & locus of points such that two separate solution curves are tangent at each point m £&& locus. Envelopes After due deliberation, it would certainly appear that not every function f(x,y,c) would have an envelope for its family of curves. Therefore some conditions for 8 an envelope will be given, first, a necessary condition for the existence of an envelope will toe stated# Theorem 2,1. If f(x,y,c) « 0, where f(x,y,c) is twice differentlable, Ml §S envelope E(x,y), then the points of the envelope must satisfy f(x,y,c) « 0 and fe(x,y»c) « 0. Proof. Let f(x,y,c) * 0 he the equation of the family of curves and E(x,y) » 0 he a curve tangent to each curve of f(x,ytc) at some point £»# Also each point of E(x,y) is a point P_ at which some curve of f(x,y,c) touches E(x,y). By Definition 2.2, E(x,y) is an envelope of f(x,yte). lake e as a parameter aid write the equation E(x,y) « 0 in parametric form. Tljua, x « x(c) : and y » y(c). From the above relations, the slope of the envelope at any point is (1) m &Z/M* dx dx/dc Next, taking the total derivative of f(x,y,c) with respect to c, (2) fvdx . f_dy ^ f„ n dc I ^dc c » 0. ! Taking the total derivative of f(x»y,c) with respect to x, (?) f + f x + °iS * °- Let fU.y.cp be oae the curves of the family fU.y.c). Therefor© c^ la a constant and equation (3) becomes w fx + fyg - 0 which holds for the single curve f (x,y ,0-^) - 0. This gives the slope at any point on f(x,y,c^) « Of and at any point P„ on this curve, it should give th© slope of c E(x,y) » 0 "by the hypothesis# It has been shown that the envelope should satisfy both equation (1) and equation (4). Eliminating dy/dx from these two relations, the following relation is ob- tained : (5) £Jll 4. C&E , o xdc + ^dc °* low it has been shown that both equations, (2) and (5), are satisfied by the point Pc on the envelope* Com- paring these two relations, it must be concluded that f *> 0 when evaluated at the point P_. Thus the theorem C V is proved. The last theorem indicates that an envelope can be found by obtaining the c-discriminant by Definition 2,1, that is, if an envelope exists. If the existence of an envelope is overlooked, it might be expected that the c-discriminant will always yield an envelope. However, this is not al?/ays true, for example, the family of curves x 2 represented by the function f(x,y,c) « jJ - (x - c) « 0 has for its c-discriminant the relation y^ » 0« Since this is a straight line with a slope of zero, f(x,ytc) would at some point have to have a slope of zero. xo However, upon examining - (x - c)^ a 0t it is found that a slope of zero does not exist at any point on the family. Therefore it is necessary that a sufficiency condi- tion he stated for the existence of an envelop©. Theorem 2.2, jjf f(x,y,c) « 0* fc(x,y,c) « 0, / 0, where f(x,y,c) is fcc(x,y,e) t 0, and fx fy fe x fc y twice differentlable, then there exists an envelope of the function f(x,y,c). Proof. Since the Jacobian of f(xty,c) » 0 and f (x,y,c) * 0 with respect to x and y is not zero, then x and y are functions of c where c is considered to be inde- pendent . By Theorem 1»4S both f(x,yfc) » 0 and fc(x,y»c) » 0 can be written in parametric form, that is, (6) x » x(c) and y ® y(c). Since f(x,y,c) is twice differentiable, fc(x,y,c) is dif- ferent iable and x « x(c) and y • y(c) have first deriva- tives. Taking the total derivative of f(xtysc) with respect to c, (7) f*tl+ M?+ f° - °- Next, taking the total derivative of f_(x,y,c) with re- V spect to c, (8) f_._dx , f_„dy , cxzs + cm + cc* A° * 11 Since f (x,y,c) / 0, dx/dc and dy/dc cannot both be cc equal to zero* Hence the curve x « x(c) and y » y(c) has a slope equal to the ratio of dx/dc and dy/dc, that is, (9) dx dx/dc Since fc(x,ytc) - 0, equation (?) becomes (10) f„dx . f rd,v n xd£ * ydc " °* faking the total differential of f(x,y,c) « 0, (11) fxdx + fydy + fcdc « 0, Holding c constant, dc « 0. Thus for any one curve of the family, (12) fxdx + fydy » 0, Since the Jacobian is unequal to zero, fx(x,y,c) and f (x,y,c) cannot both be equal to zero, and the above y equation determines the slope of the curve of the family with c held constant* Since f (x,y,c) and f (x,y,c) cannot both be zero and 5E Jf dx/dc and dy/dc cannot both be zero, it follows from equa- tion (10) that if dx/dc » 0 and dy/dc £ 0, then fx(x,y,c) / 0 and f (x»y,c) - 0. Also if dx/dc / 0 and dy/dc « 0, then y fx(x,y,c) » 0 and f (x,y,c) / 0* Comparing equations (10) and (12) after solving both relations for ~f /f > a y (15) &Z « dx dx/dc which holds for any point on f(x,y,c) « 0. 12 Therefore, comparing equation (9) and equation (13)» the curve f(x,ytc) » 0 and the curve x « x(c) and y • y(c) have the same direction at the same point. Hence the curve x « x(c) and j « y(c) is an envelope of f(x,y,c). It should be noted that the condition f„_(xty,c) / 0 Cv found in the above theorem is stated only to insure that - dx/dc and dy/dc are not "both zero. Thus there may he an envelope of f(x,y,c) when f„_(x,y»c) « 0 and dx/dc and/or dy/dc is unequal to zero. Consider, for example, the function f(x,y,c) « (x-c)M + y O - 1 » C. In this case fcc(x»y»c) " 0 when x » c and the c-discriminant is found p to be y » 1. The function f(x,y,c) describes a family of circles with centers on the x~axis and having a radius of one. Thus it can be seen that y ® 1 and y • -1 will be envelopes of the family of curves. Conditions for Some Loci Before investigating the involvement of the node and cusp loci with the c-discriminant, some conditions for the existence of these loci must be established. Node-Locus A necessary condition for the existence of a node- locus will first be given. Theorem 2.3. If f(x,y»c) » 0 has a node-locus, then §M Point (£,ti,y) on the node-locus will satisfy f(x,y,c) » 0, fx(x»y,c) - 0, f (x,y,c) « 0$ and f (x,y,c) *» 0. 13 Proof, From Definition 2.4, ©very point on the node- locus is a double point, that is, at each point on the node-locus, f(x,y,c) « 0, fY(x,y,c) » 0, and f (x,y,c) » 0. A, J Taking the total derivative of f(x,y»c) » 0 with re- spect to c, £ dx f dj f m 0. xdS + ydS + c Since f (x,y,c) « 0 and f (x,ytc) • 0, it follows that x y fc(x,y»c) • 0* Although it will not be of any immediate benefit in furthering the investigation of the node-locus, some in- terest should be taken in determining the direction of the node-locus at any point. Thus the slope may be determined by the conditions of the following theorem: Theorem 2.4« If f(x,y,c) «• 0 has a node-locus, then the direction of the locus at the point (%*t|,y) on the locus is given b£ either £ £ - f_ f- f f. "£fvY fr£*i1 *IY 4£'v 4Y £Y £FF£ ,yy * f f ~ff ff -ff nr 4y w ^cyy ty ny or f t £ - f f TiTi yy nY ny Proof* Let (4»ti»y) be one point on the node-locus and (£+d£,^+dT},Y+^Y) be another point on the node-locus. From Theorem 2.5, any point on the node-locus will satisfy (14) f(x,y,c) - 0, (15) fx(x,y,c) - 0, (16) fy(x,y,c) « 0, (1?) f (x,ytc) - 0, 14 Taking the total differentials of equations (15)• (16), and (1?), the followiny g relations are obtained*. (18) fax 4* + fwic « 0, XX vy xc f dx + (19) xy 4, v fy c„d c #8 0 , y (20) fx c dx 4- V + f cc 0C m G« At (4,iuy)* equation© (18) , (19), and (20) become (21) + * f£ydY * °t (22) f5i|aS + + , 0, (25) f5Td? + f1YST1 * fYYSY " °* Eliminating Qy from any two of equations (21), (22), and (23) * d?i/d£t which is the direction of the node-locus at (^*n*Y)» is determined. Thus, from (21) and (22), In M4 f^f ny £t] 4Y nr Equations (21) and (23) result in 4H _ - fyy£tt< dl " f f „ - f f yy 4n Cry tiy And from equations (22) and (23), on f ...f . - f„ f \dz " f~:r -""fzj ~Tin~YYvv ~ny~r\y Since the coordinates of the node depend upon c, by Theorem 1,3 d(f ,f ,f ) =8 Q must hold to insure consist- T]» Y ) ency. 15 Cusp-Locus In examining the cusp-locus» it will first be neces- sary to state a necessary condition for a function to have a cusp-locus. Theorem 2,5. £f f(xfy,c) has a cusp-locus, then any point on the CUSP-locus will satisfy f(x,y,c) « 0, *x(x»y»c) » 0, fy(x,y,c) - 0, fc(x,y,c) - 0, and f f - f 2 « 0. xx yy acgr Proof, By Definition 2.5, each point of the cusp- locus is a double point. Since (£»n»Y) is a point on the cusp-locus and a double point, it follows from Theorem 2.5 that fx(x,y,c) » 0, fy(x,y,c) » 0, and fc(x,y,c) « 0 at O (Si"n »Y) . It remains to be shown that f^f^ - f » 0 at (4»n»T>. Expanding f(x,y,c) by Theorem 1.5 for values of x, yf and c near the cusp point (£,»*} *Y)» f(ac»y»c) » f(C»T]»Y) + fx(^»n»Y)(x-§) + fy(^,Tify)(y-Ti) + (<»TI,Y)C C—Y ) + 1/2 fxx(xl'yi»cl)(x-^2 + + fcc(xl^ x »cx^c"*Y ) + ^f^Cx^^y-^, c^)(x—^)(y-ri) + 2fxc(xlfyl1cl)(x-5)(c-Y) + 2fyc(x1,y1^c1)Cy-.ti)Cc-Y) where (x^.y^c-^) is some point on the cusp-locus between the points (x,ytc) and (4, TI,Y) • When c * y and since f(2C,y*c) - 0, fx(x,y,c) » 0, fy(x,y,c) - 0, and fc(x,y,c) « 0 at the cusp point (^*TI»Y)» the above expansion becomes 16 2 f(x,y,c) - l/2[f3n[(*1,y1,c1)(x-5) + ar;1[y(x1,y1,c:L>(x-0(y-T|) + fyyC^.y-j^.Cj^Xy-n)2]. Now let x-s * Ax and y~ri - Ay. Upon substituting these values in the above relation, 3 O 3CX X ' *1 ^ X ^ + ^xy^xlt^rl|Cl^'x^r + ^yy(xi>yi»ci)^ •* 0. p Dividing by Ax and then solving for Ay/Ax, ***£ 4* f f f Ax f 3T The slop© of f(x,y,c) at the cusp point is given by Ay/Ax, Since there is only one direction at the cusp, the term within the radical must be zero, that is, ^3QT " ^xx^yy * °* Corollary 2.1. If f(x,y,c) has a cusp-locus. then the direction of the function at the cusp point is given b^ Ay/Ax - -f^f . Prpof* It follows from Theorem 2.5 that Ay/Ax » -f^/f^, since f ^ - f f » o. xy xx yy * Another property of the cusp-locus to be determined is the direction of the locus at the point (£»ti,y)» Remark 2.1. To determine the direction of the cusp- iocus at the Eginfc (E>tn%y). From the necessary conditions of a cusp-locus, (24) f«V, - - 0 at the point #tj,y) on the cusp-locus. Let G be a 1? function of (x,y,c) such that at the point (£,ri,y) 2 s(5,h,t) = tutm - f5n - o. Taking the total differential of G, (25) + G^dt) + G^dy * 0. Since the cusp point is a double point, equations (21), (22), and (23) from Theorem 2,4- hold at the point (S»n»Y>» (21) a 0 * W1 + VY (22) « 0 fSna? + V + VY (23) + f dy « 0 f6Y35 + YY r Eliminating iy from equation (25) and any one of the equations (21), (22), and (23), the direction of the cusp- locus at (C*T!»Y)* d-q/d£, is obtained. Some additional properties of the cusp-locus must be stated before beginning a study of the cusp-locus as re- lated to the c-discriminant. gheoreia 2.6. If f(x,y,c) has a cusp-locus. then the .following: relations hold at the point (£,ti,y) on the cusp- locus : f + f_. f„ 4- f + f f + f _££str _ £l . ^Tliror, + rlY . I£Y * 1YY. Si 11 11 f1Y aa& frr + f, f„ + f Sn ftiy Proof. It was shown in Remark 2.1 that equations (21), (22), (23), and (24) hold at the point (£ , y) on the cusp-locus. Thus, 18 (21) » 0 f55'5 + V* + f5YdY (22) V4 + V + VY a* 0 (25) fg/5 + f„yai. + fYySy m 0 (24) f55fni " f4>!2 " °* Eliminating di\ from equations (21) and (22), (26) (f^ - f^2)d5 + . f^fw)aY . o is obtained. Comparing equations (24) and (26), (2?) f_ f - f_ f a 0. in nr itnn The next relation is obtained by combining equations (24) and (27) which proves part of the conclusion. (28) f + f f 4. f m -i2L-ljSuc*' f V f £tl 71T1 Next, eliminating 3t| from equations (23) and (22), <29) ( )i? + C 2 )aY Wiy - Vn V - VYY - Comparing equations (27) and (29), (50) V* - fnnfYv " 0 is obtained. Then upon combining equations (27) and (50), (31) * fiY . f£r liix- r w A third relation is obtained by comparing equations (28) and (31) • that is, C52) * f^Y . iz_L£lX. f w 19 From equations (28), (31)* and (32), the following proportions are obtained which are in a convenient form for obtaining various relations involving second deriva- tives of f(x,y,c): « k-j^f n k«fff 2 " VSY » ki f Cr 1 TY % 2 T}T1 m • k3f1Y m k £ f c,1\ %1 i\y 2 TJY nr " VYY factors of the c-Discriminant As stated in Definition 2.1, the c-discriminant is the relation resulting from the elimination of the c term between f(x,y,c) « 0 and f (x,y,c) « 0* To investigate the envelope, node-locus» and cusp-locus as factors of the c-discriminant, it will be necessary at this time to es- tablish a general form of the c-discriminant. Here the function f(x,y,c) is a polynomial in c. Definition 2.7. The general form Q£ the c-discrimi- 1 Baatf. J& 2 « A®*" f(x,y,ri)f(x,y,r2) ... f (x,y#rm->1), where r r r ,37 l* 2' J» *** a-l SM roots of fc(x,y,c) - 0 and A is a function of x and y. Envelope The general method for obtaining the envelope is to find the c-discriminant. However, it has been shown that the c-discriminant does not always contain an envelope« It would be of interest then to find out how the envelope 20 is related to the c-discriminant bo that upon inspection the discriminant would indicate the presence or absence of an envelope. Therefore, it will be shown that the c-dis- criminant will, in general, contain the envelope as a factor once and only once. Theorem 2.7* If (§,n) are the coordinates of any .point on the envelope of the curves f(xfy»c) « 0t and if (x»7) M set? equal to in the c~discriminant of f(x*y,c), then this discriminant will vanishi and conse- quently this discriminant will in general contain the SKSSlS^-lSSSS om@» and only once, as a factor. Proof. from Definition 2.7, the c-discriminant is Tfi 1 S-V f(x,y,c1)f(x,y,c2) ... f (x,y,cja^1). Letting m 1 q - Am" f(x,y,c2)f(x,y,c3) ... f (x.y.c^), the discrimi- nant is S « Qf(x,y,c^). Since (4,ti) is a point on the envelope, it is known that the two equations f(4,T},c) « 0 and r(£,i],c) « 0 are satisfied by a common value of c, say cx» Thus when (x,y) is equal to (£,t0, the parameter is c^ and f (£,11,0^) « 0 and - 0. Therefore, when (x,y) is equal to a « Qf(4i^»c1). Sine© fCC^fC-j^) - 0, 2 « 0. Thus the discriminant vanishes when (x,y) is set equal to any point (C»n) on the envelope. Hence the discriminant must contain the envelope-locus as a factor at least once. Taking the partial derivative of the discriminant with respect to x, 21 6S Qdf(x,y,c,) f(xtytc, )6fi 6x 35 — + 1 6x* ox Since f(x,y,c ) = 0 when x « Z, and y » n, the above becomes 62 bm Qdf (x,y,c-, ) f.-T °x ax Similarly, it can be shown that 6g . QMCx.y.Cj) 6y g- If 6S/6x and/or 6S/6y does not vanish when x « £ and y «* Tjt then the discriminant contains the envelope-locus once and only once since all other factors of the discrimi- nant are the factors of Q» Let 62 6©H j Similarly, It can be shown that 6S/6y £ 0 if the envelope- locus is a factor only once. If n is greater than one, then 6S . e>g vn~xm- 6x bx * 6x And since 2*0, 5E/&X « 0, 5S/6y » 0, when x » § and y * tjf and 2 will in general contain the node-locus twice, and twice only, as a factor* Proof* If (£»*i) is a point on the node-locus, then x a y a t], and c « c^x^y), where Cj(5fn) « y, will satisfy the equations f(x,y,c) « 0, f (x,y,c) « 0, *B?L- fy(x,y,c) * 0, and fc(x,y,c) « 0. It was shown in Theorem 2.7 that E - Qf(x,y,c1). faking the derivative of S with respect to x and then y, £3 « Q*v(x«y»Ci) + f(x»yic, )6g &x x x 1 6x |£ „ Qf (x^y,^) + f(x,y,c1)6fi. fi"sr « Since f(x,y,c1) * 0, f^Cxjj,^) « 0, and fy(x,y,c1) « 0, the preceding equations become 5c " 0 &.S n Since E » 0 at (^, rt), it follows that the discrimi- nant 2 contains the node-locus as a factor at least once low let m m m . m Since tp • 0 and 62/6x - 0, the last equation .becomes Since 6 62g 2 611I,JI1 5 26R 6<£. ' 11J JK muaum mmmrn &x2 6x 6x In general, 68/6x and 6 cusp-locus will occur in the c-di3criminant4 in general, as a factor three times and only three times# Theorem 2.9, If f(x,y,c) has a cusp-locus and (£,T) ,Y) is a point on the cusp-locus« then the c-discrlminant $ 2 2 2 2 2 65/6x, bey by, 6 S/6x s 6 3/6y , and 6 S/6x6y will all vanish when x » ^ y « ri, and e-^ » Yf and the c-discrimi- nant will in general contain the cusp-locus three times„ and only three times, as a factor. Proof. From Theorems 2.5 and 2.6, the following re- lations hold at the point (£•?]»Y) on the cusp-locus: (54) f • 0, fx * 0, fy - Of fc - 0, ^ and (55) f££ •» f? f- + f f- + f £lj t^Tj T1Y Since the conditions of (34) are the same for both the node-locus and the cusp-locusr it can "be shorn that S - 0, bZ/bx - 0, and 6£/6y » 0 will hold at (£,tuy) for the cusp-locus just as they did for the node-locus in Theorem 2,8. From Theorem 2.8, the second partial derivative of S withp respec t to x is given as &-§ . 3[*xx(x.y.c1) + fxc (x,y,c1)^]L]+ 2fx(x,ytc1)6^ w*- l 1 J 6x f (:x y c ) + f + c1 ' * l ||lg (x»y,c1)£^. 6x " 26 From the conditions of (3#)» the last relation becomes e 2r-» (36) _6 a Q f5aC(x,y»c1) - fX(J (xsy,c1) toe 3* f- - (x^c,) g1C1 By evaluating equation (36) at (^• "nand applying the conditions of (53), equation (36) "becomes 622/f>52 - 0. Similarly, it can "be shown that 62E/6r\2 * 0 and . 2 6 £/6^6t] » 0. Thus the discriminant S contains the cusp- locus as a factor at least twice. O If 6>.o;. ^ < P+ ,2 S 2® 2 ft 2 2 6 "v£j 2S 6x 6x Since b Sine© S « 0, the last relation "becomes 6 _ "68. 6x5 83C In general 5 Thus it is concluded that, in general, the c-discrimi- nant will contain the cusp-locus three times, and only three times, as a factor. CHAPTER III THE p-DISCHIMINANT LOCI Although the family of curves discussed in the past chapter was not necessarily a solution of a differential equation, it certainly can represent the general solution of a differential equation. Thus, if f(x,y,c) « 0 is the solution of ?(x,y»p) « 0, then it has already been shown that the c-discriminant will contain the envelope, the node-locus, and/or the cusp-locus if any of these loci exist. Therefore it was through the function of the solu- tion, f(x,y,c), that the various loci of F(x,y,p) were determined. It would seem that some of these results could "be obtained through an examination of the differ- ential equation itself. There is a method which is very similar to the method used with the solution function. That is, find the p-discriminant by eliminating p between F(x,y,p) ~ 0 and F (x,y,p) » 0. This method is used to Jr find the envelope and it also is found to reveal the cusp- locus and the tac-locus, but not the node-locus. It is noticed that finding the p-discriminant is much the same as finding the c-discriminant, with the only ap- parent difference being in p and c. This would lead one to believe that the proofs for the conditions of an envelope OO 29 of f(x,y,c) in Chapter II would suffice for proving the conditions of an envelope of F(x,y,p) simply by inserting p in place of c. However this is not the case since c has a constant value for one curve of the family and certainly p takes on several values for this same curve. Difficul- ties arise in attempting to state both necessary and sufficient conditions for an envelope of F(x,y,p). There- fore, in this paper only a necessary condition for the envelope of the differential equation .F(x,y,p) will he given. Lemma 3.1. If f(x,y,c) » 0 and by the implicit func- tion theorem may be written as y » G(x,c) and if fc(x,y»c) « 0, then &c(x,c) = 0. Proof. Taking the derivative of f(x,y,c) with respect to c while holding x constant, ^.dy/dc + fc * 0. Upon rearranging this, (1) §2. _ l£c. K de f y •Taking the derivative of y «= G(x,c) with respect to c while holding x constant, (2) §2.x Ot (x»c). dc Eliminating dy/dc between equations (1) and (2), (3) % (x,c) « ~£c. If an envelop© of f(x,y,c) exists, then by Theorem 2.1 fc(x,y,c) = 0. Thus it is seen from equation (3) that if an envelope exists, then Gc(x,c) « 0* 30 Theorem 3.1. If the differential equation l(x»y,p) has an envelope, then each •point on the envelope must satisfy F(x,y,p) • 0 and, F (x,y,p) » 0, Proof* First consider the differential equation (4) F(x,y,p) « 0 whose general solution consists of a family of curves (5) f(x,y,c) « 0. By Theorem 1.1, equation (5) may be written in the form (8) y » G(x,c) where G is a function of the independent variables x and c. Taking the derivative of y with respect to x, (?) ^ Gx(x,c). dx Since p » dy/dxt equation (7) becomes (8) p « Gx(x,c). If equation (8) were solved for c, the results would be c » j » G£x, (9) &2L _ Gc(x,c)dc. ap - ° aP This value dy/dp holds for any point on f(x,y,c) - 0 or y - G(x,c). Taking the derivative of (4-) with respect to p, (10) F dx F dy F ^ J x + + p « 0. Since x and p are independent variables, (10) becomes 51 (11) Fyg + fP - o. Eliminating dy/dp between (9) and (11)» 5" ^(x»c)dc 4- * 0. J dp If an envelope exists, G (x,c) « 0 by Lemma 3*1* There- v for©, if an envelope exists, P « 0, P Loci of Contacts of Parallel Tangents To investigate the envelope and extraneous loci as- sociated with the p-discriminant, it will be necessary to entertain the idea of loci of contacts of parallel tangents. These loci are determined by the points on the curves of the family having equal values of p. then considering p as a constant, F(x,y,p) behaves in the same manner as did f(x,y,c) in Chapter II. To avoid confusion, a can be used as the parameter when p is a constant# Thus F(xfy,p) can be represented by F(x,y,a) when p is the constant a. It will first be shown that the loci of contacts of parallel tangents will touch the envelope-locus. That is, there will be at least one point where the direction of the loci of contacts of parallel tangents is the same as the direction of the envelope-locus. Theorem 3.2, If f(x,y,c) has an envelope. the point (StU) ia on th£ envelope, and (S,tj) is a point on the con- tacts of parallel tangents. then the locus of contacts of 52 touches the envelope-locus. That is, both loci have the same direction at (£,rj)« Proof* Let a be the angle between the x-axis and the tangent to f(x,y,y) = 0 at (£Un) where is a point on the envelope and y is a constant parameter. If the parameter of f(x,y,c) « 0 is a constant, say y, then it has been shown in Theorem 2.1 that the slope of the curve at any point (x,y) is equal to -f ^/f^. Thus for the curve f(x,y,y) « 0, tan a » -f /f when x « £ and y » t). X ,y Similarly, for the curve f(xsy,y+dy) « 0, tan a * -f^/fy when x » and y « n+dtj. Since f(£>d^fT!*dtj%y+dy) » 0 and f(^»ti,y) « 0, the total differential of f is equal to aero, that is, (14) 3§3f x 0T]df i dydf A a| + + ay " 0 Since ( £ , 7} ) is a point on the envelope, then by Theorem 2.1 df/dy « 0. Thus equation (14-) becomes (135 KM. + ®t|df m 0 af + di * °* Solving equation (15) for dri/d£, which gives the slop© of the loci of contacts of parallel tangents, la -a*/ag. 48 ~af75i Therefore it is found that the direction of the locus of contacts of parallel tangents is equal to -t /f , It has previously been shown that this ie also the direction of the envelope of f(x,y,c) « o at (4, ti) . Thus the loci 33 of contacts of parallel tangents touch the envelope. Also all points on the locus of contacts of parallel tangents satisfy E(x,y,a) * 0 where a « tan a. Next it will he shown that the loci of contacts of parallel tangents touch the cusp-locus of f(xty,c). Theorem 3»3- If f(x,y,c) » 0 has a cusp-locus, then the loci of contacts of parallel tangents touch the cusp- locus. Proof, If (4,ti,y) is a point on the cusp-locus of the function f(x,y,c) » 0, then by Theorem 2«5» f„ « 0, f « 0, and f « 0 at Also it has been shown in y ° Corollary 2.1 that the slope of f(xty,c) at the cusp point (£»HiY) is given by -f^/f^. Thus the slope at (£,tj,y) is Let (£+d£»1l+(Jll»Y+ciY) be a point where the tangent is parallel to the tangent at (£,twy)• Since the slope at any point on f(x#y,Y+dy) is given by -f^fy, the slope at this point is -f^/fy where x « E.+ &E, and y » -n+d-n. Since the tangents are parallel, -f^/f^ - -f^fy where x » 4+dg, and y « Expanding f and f by M, y Theorem 1.5, Taylor's Theorem, the following results are obtained; (16) tt * ttlae. + fgT|a»i * f£TaY * 1/2 b f + + f dT + 1/,a 3 >i „Y 2 2 2f s d where E - + f^Sr, + f§YY«T + ^„ S 1 + 2fWYe?#T t 2f4w 34 2 2 aad a . • fw • 2ft,T«5*r - 2f„w»i«Tf. From Remark 2.1, » 0 and fe d£ + f dtj + fm 3y » 0. Therefore, equation (16) T)T} Tiy becomes H - Sf^ « 0. f By substituting the values of B and S found above In tiiis relation, the following equation is obtained; 'm?*5* + W2 + Hrx*? + "wi*44" + zs^Kiy • 2tmWy - szutlm*toi - - "aW* hn fei hn - - 'aW* ~ £i&a£l . o. f4n £^n Let A, B, C, D, £, and F take on the following values: A " f5?5 " %f£Sn' B " ' £g§5 f5n 0 * f5YY " fSSft|YY' B " f?HY ~ r5n f5i E * f44Y " f££f£iY' * * f?5l " il3? Thus the above equation becomes, after the substitution of A, B, C, D, 1, and f, (1?) Ad£2 + B&n + G&r + 2DdTidy + 2E3§dy + 2Fd£dr] « 0, 35 This relation is satisfied by any point on the locus of contacts of parallel tangents. Since (£,"n) is a point on the curve f(x,y,y) and (£+34,11+3x1) is a point on f(x,y,y+dy), f (£,t),y) « 0 and f(4+d4>-n+dT|tY+dy) « 0. Expanding f (£+$£, ti+^ti ,y+dy) » O by Theorem 1.5» the following relation is obtained; 2 d 2 + f^U.Twy^n + £yyU^\>v) y + 2f^(4,t|,Y)d^dii + 2f^(£,Ti,Y) By theorem 2.6, - f^ - 0, - f^2 - 0, and f$5fny " f4nf£y 0 hold at the cusp point (&*tl»Y)« Using these conditions and by multiplying by f^t the above equation becomes + ^n2""2 + VV + + 2f^f4Yd^dy + 2f^tif^Y * °* By factoring, the above relation becomes (IB) f^dc, + f^dn + f^dy » 0. Equation (18) is satisfied by any point on the locus of contacts of parallel tangents. The direction of the locus of contacts of parallel tangents is obtained by eliminating dy from (17) said (18)* Thus from (18), dy « -f^d^ - f^dn and upon substituting f~ ' Cy this in (1?), the following is obtained after some simpli- fication: 36 2 2 2 (19) (Af , + Cf - 2Bf,FfF)dE> 2 + (2Cf^f^ - 2Df^Y - 2!f^f^Y + 2ff^Y )d4dn 2 2 2 + (Bf^ + Cf^ - 2Df^f4Y)dn » 0. In equation (19) let L be the coefficient of «J£2 ? the coefficient of d^dri, and H the coefficient of dip. This results in the equation Ld£2 + Md^dn + Sdrj2 « 0. Solving this relation for t)£/dri t 3£ -M - ^1T~ 4U, N 2L Since the locus has only one direction at (£»tj»y)» I2 - 4LN - 0, Therefore, a^/dt) « -M/2L or Ld£ + HdT) « 0. 5 Inserting the values of L and M, the above relation gives the direction of the loeus of contacts of parallel tangents at the point (£,twy)« 2 2 (20) (Af^Y - 2Eff^f^Y + 0f^ )d^ Df f Ef f Ff 2 dT + (Cf^f^ - ^ ^n ~ 4T) 4Y + ^y ) i - She conditions at the cusp point (§*tuy)> (21) £t f£ •i.nni»i*d»—<1 r*. w.mfrwJh msh immwL "S ® m f m W tiy are given by Theorem 2,6. Thus the relations for the cusp at (g+d^ti+diOt when c « Y+dY» is obtained by taking the total differential of (21), That is, 37 £-££ a « -qJLI) TH T)y where D represents QEO + drid_ + dyd_. The above relation dy •becomes fy DfSJf - fpg-Df- f Dfr - 1' D£ JX , C,n ^,-n B Tin £n En nn f- f 2 tjtj f T)f - f Df p f ^ mT Performing the operation as indicated by D, 1jmK + i H + fMT>T _ f4l 1 2 . f^Sia4 * f^iaT| * f55Y„''T _ fs/£T}n"5 * f£nfniiTiaT| * fEnfrniv8T TJTi . kzxlLllmllJJml1_ * ',vWy. X p ny f*ny A better form of this relation is obtained by factoring out the d£, dt}4 and dy terms. Doing this, the following is obtained; (f f f f £££ £B ~ + < ££B £n ' ^EE^Etiti^ fr f ^ (frs fy - Y *" . i_mcni! ^reriTi- 2 r • f%T| f f f aT1 f f f dY + ^- nn £-nT) ~ ^T) nTiin^ + ^ E&Y Ti7> *" gAirP „ ^f^^yfTiY " ff^nYfT)Y ,~. f ag^nar^I! m A ~ O i> J? £•» T)Y T1Y (f_ f - f- f )dy. + , c,YY nx, , 6y nyy ,.: 2 friy Letting A, B, 0, D, 1, and F have the same meaning as stat- ed earlier, this last equation can be written in the simpler form of (22) Ad£ + Fdy + Bdy ffdf, * Bdn • Mr Ml + * COy. fim ^tiy Equation (22) is therefore satisfied by the point (4,n»Y) on the cusp-locus. from Theorem 2.6, the points on the cusp-locus satisfy (23) f^d£ + f^dr) * f^yay . 0. The direction of the cusp-locus may he determined by eliminating dy from (22) and (23), Using the first and third relations in (22) and with equation (25) ? the direc- tion of the cusp-locus at (£»tj»y) is given by 2 2 (24) (Af^y - 2Ef^f^ + Cf^ )a^ 2 (Of^f^ - Df^^f.Y -fif^^f ^Y+ Ff4Y )dn » o. Comparing equations (20) and (24), it is seen that the directions of the cusp-locus and the locus of contact points of parallel tangents are the same at their common point (4tn,y). Thus it is concluded that the loci of con- tacts of parallel tangents touch the cusp-locus. 39 In order to find some relation between the tac-locus and. the p-discriminant, it will first be shown that the tac-locus is a part of the node-locus of the loci of con- tact points of parallel tangents of f(x,y,c) « 0. Theorem 3,4* If f(x,y,c) « 0 has a tac-locus, then the tac-locus is< in general« a part of the node-locus of the loci of contacts of parallel tangents to the family of curves f(x,y,c) « 0. Proof. Let (£,*]) he a point on the tac-locus such that it will satisfy both relations, f(5»,Tj»Y^) * 0 and She slope at any point on the curve f(x*y*Y^) « 0 is given by dy/dx « -f^fy. Thus at (£,t|) the slope is -f^(^,T],y1)/f^(^1rijy^). Similarly, the slop© ;f T of f(x,y,Y2) » 0 at (4,ti) is -f^(^,n,Y2)/ '71(C» l»Y2)• The slopes of both curves at (4*1) are equal, thus Since -f^fy is the slope at any point on a given curve of the family, let tan a be the direction at the points (x,y»Y]_) and (x,y,Y2). Thus the slope of the curve at any point on the locus of contacts of parallel tangents is equal to tan a where tan a « -f^/fy. Since -f^fy remains constant, take its total differ- ential and obtain x d w d "fxl f f. f„ y I JJ * dx dx + Qy dy + dc dc 0. 4-0 faking the total differential of f(x,y,c) « 0, df, A df. ^ if a n + ^dy + -^ac « 0. Eliminating dc from the last two equations, a. f"S£W - M£VV dx * TTWiZJiZT "' c-g^ r y' - y*^ x y which gives the direction of the locus of contacts of parallel tangents. Thus, at some point (£,t) ,c), an f«i£VV - fcf£W, a5 ' 'eftV•an * V'' - *riw'oc The direction of the locus of contacts of parallel tangents will, in general, be different when c is given different values. That is, when c » a different slop© is obtained than when c » y^. Thus at a point (£,t]) on the tac-locus, the locus of contacts of parallel tangents has two directions* This indicates that the point on the tac-locus is also a nodal point on the locus of contacts of parallel tangents, Thus it is concluded that the tac-locus is at least a part of the node-locus of the contacts of parallel tangents. Factors of the p-Dlscriminant The last three theorems will enable one to now show the relations between the envelope, cusp-locus, tac-locus, and the p-discriminant. It will be shown, in general, that the p-discriminant will contain the envelope-locus as a 41 factor once, the cusp-locus as a factor once, and the tac- locus as a factor twice. Theorem 3«5* If the differential equation F(x,y,p) * 0 has for its solution the family of curves f(x,y,c) « 0 and if an envelope. cusp-locus, and tac-locua exist for f(x,y,c) « 0, then the p-discriminant contains. in R-eneral. the envelope-locus as a factor once, the CUSP-locus as a factor once. and the tac-locua twice as a factor. Proof. Since F(x,y,p) « 0 is the relation represent- ing the differential equation of the family of solution curves f(x»y,c) » 0, F(x,y,a) » 0 represents the loci of contacts of parallel tangents. It was stated in Theorem 3«2 that the loci of contacts of parallel tangents touch the envelope of the curves f(x,y,c) - 0. Therefore F(x,y,a) • 0 contains the envelope as a factor. Since a remains con- stant for a given locus of points of contacts of parallel tangents, F(x,y,a) > 0 may be treated in the same manner as f(x,y,c) » 0 was treated. That is, the a-discriminant would be found in the same manner and it is concluded that the a-discriainant will, in general, contain the envelope- locus once and only once. Now since a represents the slope at a point on the family, it is actually an arbitrary value of p. Therefore, the a-discriminant is the same as the p- discriminant« Thus it is concluded that the p-discriminant will, in general, contain the envelope-locus once and only once, 42 Similarly, it has "been shown that F(x,y,a) » 0 touches the cusp-locus. Therefore, the cusp-locus is a part of the envelope of the curves F(x,y,a) * 0. Aa above, the a-dis- criminant contains, in general, the envelope locus once and only once# Since the cusp-locus of f(x,y,c) « 0 is a part of the envelope of F(x,y,a) • 0, the a-diacriminant will, in general, contain the cusp-locus once and only once. Thus it is concluded that the p-discriminant will, in gen- eral, contain the cusp-locus once and only once. It has been shown that the tac-locus of the curves f(x,y,c) » 0 is, in general, a part of the node-locus of the contacts of parallel tangents, F(x,y,a) « 0. Since the a-discri.rolnant has been shorn to contain, in general, the node-locus as a factor twice and only twice, it is con- cluded that the p-discriminant will, in general, contain the tac-locus as a factor twice and only twice. BIBLIOGRAPHY Books Courant, R«, Differential and Integral Calculus, Vol, II, lew York7 Interscience Publishers, IncT, 1936. Franklin, Philip, Methods of Advanced Calculus % New York, McGraw-Hill Book Co., Inc., 1944-. , A Treatise on Advanced Calculus, New York, John Wiley and Sons, Inc., T§407 Kells, layman M., Elementary Differential Equations. New York, McGraw-Hill Book Co., Inc., I960. Miller, Frederic H., Partial Differential Equations. New York, John Wiley and Sons, Inc., 1941. Murray, Daniel A., Introductory Course in Differential Equations. New York, Longmans, Green, and Co,, 1924. Piaggio, H. T. H., An Elementary Treatise on Differential Equations and Their Applications. London, G. Bell and Sons, Ltd., 1933. Articles Cayley, "On the Theory of the Singular Solutions of Differential Equations of the First Order," Messenger of Mathematics. II (1872), 6-12. Hill, M. J. M., **0n the c- and p-Discr iminants of Ordinary Integrable Differential Equations of the First Order," Proceedings of the London Mathematical Society, first series, XIX Tjune, 1888), 561-589. •, w0n the Singular Solutions of Ordinary Differential Equations of the First Order with Transcendental Coefficients,** Proceedings of the London Mathematical Society, second series, xVlI 1917), 149-183.