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 <iZg about zQ and a region B containing 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. <P be functions of the varia- bles x, y, z. These three functions will be functionally dependent. that is. there will exist a relation F(a, £>,<p) « 0 which does not involve explicitly any of the variables x, yt z, if and only if the Jacobian vanishes identically. That is. J * KSx&jjl * q.3 d(x,y,z) Theorem 1.4» Let there be given two functional re- lations , F(x,y,z) « 0 and G(x,y,z) » 0. Let these equa- tions be satisfied for the set of values x «• x , y « y , and z « z , and let the functions I and G and their first partial derivatives be continuous in the neighborhood of 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 » <p(z) and y « r(z) satisfying the equations I » 0 and G « 0 identically and n, '.NI.'HI'UJW, WP'M ******** 4 moh that xQ - (p(z0) and yQ - t(z0), Theorem 1.5* Let f(x,y,z) and all its partial deriv- atives up to and including those of order n be dlfferen- tiable near (a,b,c), then 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.