The Lagrange-Charpit method

THE LAGRANGE–CHARPIT METHOD∗ MANUEL DELGADO† SIAM REV.c 1997 Society for Industrial and Applied Mathematics Vol. 39, No. 2, pp. 298–304, July 1997 005 Abstract. We give a rigorous description of the Lagrange–Charpit method used to find a complete integral of a nonlinear p.d.e. adapted for a university course in differential equations. Key words. integral surface, complete integral, Pfaff’s equation AMS subject classifications. 35-01, 35F20 PII. S0036144595293534 1. Introduction. The concepts of the complete integral and the Lagrange– Charpit method are topics which appear with some frequency in texts which study nonlinear p.d.e.s in a classical way. There are some which do not use them; thus [3] and [5] describe only the method of characteristics. But the method of characteristics provides the integral surface solution of the Cauchy problem with uniqueness of solution; so, for problems without uniqueness, transversality or compatibility (see [5]) will not be true. References [1] and [7] introduce the concept of the complete integral and indicate its utility for Cauchy’s problem but don’t give a method to find it. Other texts describe the method without a detailed discussion of the hypotheses or proof which justifies it (see [8] and [9]). The most exhaustive study of this matter is in [4], but this book is not suitable for a modern course on differential equations. The objective of this note is to describe how to reach the most important results with a reasonable amount of work. We find an interesting historical review of concepts of solution for nonlinear p.d.e.s of first order in [2] and a modern view of the method of Lagrange–Charpit from the point of view of the geometrical theory of p.d.e.s in [6]. 2. The complete integral. It is well known that if a monoparametric family of integral surfaces of a p.d.e. of first order admits a real envelope, then this envelope is also an integral surface of the p.d.e. Moreover, it is easy to see that the p.d.e. satisfied by the functions implicitly defined through expressions of the form h(φ1(x, y, z),φ 2(x, y, z))=0,φ 1 ,φ 2given,h∈C 2 (R 2 ) arbitrary is a quasi-linear p.d.e. of first order. The idea of Lagrange is as follows: the solution of a nonlinear p.d.e. can’t be of the previous form, but if we know that a biparametric family of integral surfaces and both parameters are connected through an arbitrary regular function and the resulting uniparametric family has a real envelope, then this envelope must also be an integral surface of the equation. Let us consider the nonlinear p.d.e. of the first order (1) F(x, y, z, p, q)=0 ∗ Received by the editors March 16, 1995; accepted for publication (in revised form) October 3, 1995. http://www.siam.org/journals/sirev/39-2/29353.html †Differential Equations and Numerical Analysis Department, University of Seville, C/ Tarfia s/n, 41012 Seville, Spain ([email protected]). 298 Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSROOM NOTES 299 being F:˜ Uט V⊂R3×R2→R,˜ U, ˜ Vboth open, F∈C2(˜ Uט V), and F2 p+F2 q6=0 with p=∂z ∂x,q =∂z ∂y . DEFINITION 1. A complete integral is a biparametric family (2) Φ(x, y, z, a, b)with Φ∈C2(U×Λ),U⊂R 3 ,Λ⊂R 2 both open such that (3) rank ΦaΦxa Φya Φza ΦbΦxb Φyb Φzb !=2 and such that for every (a, b)∈Λthe expression Φ(x, y, z, a, b)=0 determines one or several integral surfaces z=ϕ(x, y)of (1). Hypothesis (3) is sufficient to ensure that the constants aand bare independent. THEOREM 1. Let us consider the p.d.e. (1).Let z=ϕ(x, y, a, b),ϕ∈C 2 (G×Λ),G⊂R 2 open be a family of integral surfaces determined by the complete integral Φ(x, y, z, a, b)=0. Let (x0,y 0)∈Gand a0∈Rbe such that there exist a neighborhood of a0,I, and a function ρ:I→R,ρ∈C2(I)such that (a, ρ(a)) ∈Λ∀a∈I. If the uniparametric family z=ϕ(x, y, a, ρ(a)) has a real envelope in a neighborhood of (x0,y 0,a 0), then this is an integral surface of (1). Proof. We obtain the envelope of the family z=ϕ(x, y, a, ρ(a)) by eliminating a from the system (4) (z=ϕ(x, y, a, ρ(a)), 0=ϕ a (x, y, a, ρ(a))+ϕb(x, y, a, ρ(a)) ·ρ0(a) in a neighborhood of a0,whichwealsodenoteI. The second equation of (4) determines a function, defined in a neighborhood of (x0,y 0), a=a(x, y) (with a0=a(x0,y 0)), and placing this in the first equation we obtain z=ϕ(x, y, a(x, y),ρ(a(x, y))). This surface is an integral surface because zx=∂ϕ ∂x +∂ϕ ∂a ∂a ∂x +∂ϕ ∂b ρ0∂a ∂x =∂ϕ ∂x, zy=∂ϕ ∂y +∂ϕ ∂a ∂a ∂y +∂ϕ ∂b ρ0∂a ∂y =∂ϕ ∂y , Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 300 CLASSROOM NOTES and so F(x, y, z, zx,z y)=Fx, y, ϕ, ∂ϕ ∂x,∂ϕ ∂y a=a(x,y) b=ρ(a(x,y)) =0. Remark. A sufficient condition for the existence of a real envelope is ϕaa +2ϕ abρ0+ϕbbρ02+ϕbρ00ca=a06=0. Furthermore, we can sometimes obtain from the complete integral other integral surfaces that we will call singular integral surfaces. THEOREM 2. Let z=ϕ(x, y, a, b),ϕ∈C 2 (G×Λ) be a family of integral surfaces determined by the complete integral of (1).LetP≡(x 0 ,y 0,a 0,b 0)∈G×Λand denote u0=ϕ(x0,y 0,a 0,b 0). Suppose that (i) ϕax(P)ϕby(P)6=ϕay(P)ϕbx(P), (ii) ϕaa(P)ϕbb(P)6=ϕ2 ab(P). Then there exists a unique singular integral surface defined in a neighborhood of (x0,y 0), envelope of the biparametric family, which satisfies Fp(x0,y 0,u 0,ϕ x(x 0,y 0,a 0,b 0),ϕ y(x 0,y 0,a 0,b 0))=0, F q (x 0 ,y 0,u 0,ϕ x(x 0,y 0,a 0,b 0),ϕ y(x 0,y 0,a 0,b 0))=0. Proof. Let us consider the system      u=ϕ(x, y, a, b), ϕa(x, y, a, b)=0, ϕ b (x, y, a, b)=0. Since by (ii) ∂(ϕa,ϕ b) ∂(a, b)(P)6=0, it follows that there exist two functions that we also denote aand b: a, b :G0⊂G→Λ0⊂Λ such that ϕa(x, y, a(x, y),b(x, y)) = ϕb(x, y, a(x, y),b(x, y))=0 ∀(x, y)∈G0. Substituting these values in the first equation, we get u=ϕ(x, y, a(x, y),b(x, y)) (x, y)∈G0, a surface which contains the envelope of the biparametric family. We will prove that this envelope is a singular integral surface. In fact, let ϕbe the integral surface derived from the complete integral F(x, y, ϕ(x, y, a, b),ϕ x(x, y, a, b),ϕ y(x, y, a, b))=0 (x, y, a, b)∈G×Λ. Differentiating with respect to aand b, (Fuϕa+Fpϕxa +Fqϕya =0, F u ϕ b+F p ϕ xb +Fqϕyb =0. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSROOM NOTES 301 Taking (x, y)∈G0, it follows that ϕa=ϕb= 0 and (Fpϕxa +Fqϕya =0, F p ϕ xb +Fqϕyb =0, and by (i) we will have Fp=Fq=0. 3. The Lagrange–Charpit method. We will look for a complete integral for (1) of the form Φ(x, y, z, a, b)=Ψ(x, y, z, a)−b. For every fixed b, the equivalence Φ(x, y, z, a, b)=0⇐⇒ Ψ(x, y, z, a)=b represents a uniparametric family of surfaces whose normal vector at every point is (∂Ψ ∂x ,∂Ψ ∂y ,∂Ψ ∂z ). But if we consider the explicit equations defined implicitly by the former family, z=ϕ(x, y, a, b), we have ∂Ψ ∂x ,∂Ψ ∂y ,∂Ψ ∂z =−∂Ψ ∂z (p, q, −1). This would mean that Pfaff’s equation p(x, y, z, a)dx +q(x, y, z, a)dy −dz =0 is integrable. So we must seek expressions p(x, y, z, a)andq(x, y, z, a) that satisfy (1) and that permit us to build an integrable Pfaff’s equation. We do this with the following theorem. THEOREM 3. Let (x0,y 0,z 0)∈U,(p 0,q 0)∈V,a 0∈˜ Λ⊂Ropen be such that F(x0,y 0,z 0,p 0,q 0)=0and let G:U×Vט Λ→RbeaC2function. Suppose that (5) ∃P0≡(x0,y 0,z 0,p 0,q 0,a 0)∈U×Vט Λsuch that ∂(F,G) ∂(p, q)(P0)6=0. Then there exists an open neighborhood ∆of (x0,y 0,z 0,a 0)∈R 4and two unique C1functions p, q :∆⊂R 4→Rdefined in ∆which satisfy (i) (6) F(x, y, z, p(x, y, z, a),q(x, y, z, a))=0 ∀(x, y, z, a)∈∆, G(x, y, z, p(x, y, z, a),q(x, y, z, a),a)=0 (ii) ∀(x, y, z, a)∈∆, (7) ∂(F,G) ∂(p, q)q∂p ∂z −p∂q ∂z +∂p ∂y −∂q ∂x −FpGx−FqGy−(pFp+qFq)Gz+(F x+pFz)Gp+(F y+qFz)Gq=0. Proof. By the implicit function theorem ∃ε, δ > 0,∃!p, q :B1((x0,y 0,z 0,a 0),δ)→B 2((p0,q 0),ε) Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 302 CLASSROOM NOTES of regularity C1such that (i) p(x0,y 0,z 0,a 0)=p 0 ,q(x 0 ,y 0,z 0,a 0)=q 0 , (ii) F(x, y, z, p(x, y, z, a),q(x, y, z, a))=0 ∀(x, y, z, a)∈B1≡∆, G(x, y, z, p(x, y, z, a),q(x, y, z, a),a)=0 whichis(6). If we differentiate the former equations with respect to x,        Fx+Fp ∂p ∂x +Fq ∂q ∂x =0, G x+G p ∂p ∂x +Gq ∂q ∂x =0, from which (8) FxGp−FpGx−∂(F,G) ∂(p, q) ∂q ∂x =0. Differentiating with respect to yand eliminating ∂q ∂y ,weget (9) FyGq−FqGy+∂(F,G) ∂(p, q) ∂p ∂y =0. Differentiating with respect to zand eliminating ∂p ∂z and ∂q ∂z , we obtain, respectively, (10) FzGp−FpGz−∂(F,G) ∂(p, q) ∂q ∂z =0, (11) FzGq−FqGz+∂(F,G) ∂(p, q) ∂p ∂z =0. Then, multiplying (11) by qand (10) by pand adding with (9) and (8), we obtain (7). THEOREM 4. Let G:U×Vט Λ→RbeaC2solution of the linear p.d.e. (12) FpGx+FqGy+(pFp+qFq)Gz−(Fx+pFz)Gp−(Fy+qFz)Gq=0 which verifies hypothesis (5). Let p,qbe the functions whose existence is proved in Theorem 3. Then Pfaff’s equation (13) p(x, y, z, a)dx +q(x, y, z, a)dy =dz is integrable and its general solution is a complete integral of (1). Proof. It suffices to prove that the necessary and sufficient condition for the integrability of (13) (see [8]) is (p, q, −1) ·curl (p, q, −1)=0⇐⇒ q∂p ∂z −p∂q ∂z +∂p ∂y −∂q ∂x =0 and that this condition follows from (7) and (12). Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSROOM NOTES 303 The definition of integrability implies that there exist µ:∆→R,C 1such that µ(¯x)6=0∀¯x∈∆ and a function Ψ : ∆ →Rsuch that                  ∂Ψ ∂x (x, y, z, a)=µ(x, y, z, a)p(x, y, z, a), ∂Ψ ∂y (x, y, z, a)=µ(x, y, z, a)q(x, y, z, a), ∂Ψ ∂z (x, y, z, a)=−µ(x, y, z, a). As we know, the general solution of (13) is the expression Ψ(x, y, z, a)=b. And since ∂Ψ ∂z (x, y, z, a)6= 0, the implicit function theorem allows us to define a function ϕ:B1((x0,y 0,a 0,b 0),δ 1)→B 2(z 0,δ 2), z=ϕ(x, y, a, b) such that Ψ(x, y, ϕ(x, y, a, b),a)=b∀(x, y, a, b)∈B1. Then ∂Ψ ∂x +∂Ψ ∂z ∂ϕ ∂x =0 ⇒µ(x, y, ϕ(x, y, a, b),a)p(x, y, ϕ(x, y, a, b),a)−µ(x, y, ϕ(x, y, a, b),a)∂ϕ ∂x =0 ⇒∂ϕ ∂x(x, y, a, b)=p(x, y, ϕ(x, y, a, b),a). Analogously, ∂ϕ ∂y (x, y, a, b)=q(x, y, ϕ(x, y, a, b),a). And so from (6) it follows that Fx, y, ϕ(x, y, a, b),∂ϕ ∂x(x, y, a, b),∂ϕ ∂y (x, y, a, b)=0 ∀(x, y, a, b)∈B1. Condition (3) is rank ΨaΨxa Ψya Ψza −10 0 0 ! =2 and is clearly true. Is well known that equation (12) furnishes C1first integrals of the characteristic system (14) dx Fp =dy Fq =dz pFp+qFq =dp −(Fx+pFz)=dp −(Fy+qFz). Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 304 CLASSROOM NOTES The Lagrange–Charpit method consists of (i) finding a first integral of (14) which satisfies (5), G(x, y, z, p, q)=a, (ii) obtaining pand qfrom the system (F(x, y, z, p, q)=0, G(x,y,z,p, q,a)=0, (iii) solving Pfaff’s equation p(x, y, z, a)dx +q(x, y, z, a)dy −dz =0, (iv) the expression Ψ(x, y, z, a)=bbeing a complete integral of (1). Remark. Hypothesis (5) can be weakened. It suffices to require that the Jacobian of Fand Gwith respect to two variables is not zero (see [4]). Acknowledgment. I am grateful to the referee for his kindness. REFERENCES [1] R. COURANT AND D. HILBERT,Methods of Mathematical Physics, Vol. II, John Wiley, New York, 1962. [2] S. S. DEMIDOV,The study of partial differential equations of the first order in the 18th and 19th centuries, Archive for History of Exact Sciences, 26 (1982), pp. 325–350. [3] B. EPSTEIN,Partial Differential Equations, McGraw–Hill, New York, 1962. [4] A. R. FORSYTH,Theory of Differential Equations, Dover, New York, 1959. [5] P. HARTMAN,Ordinary Differential Equations, John Wiley, New York, 1964. [6] R. HERMANN,Geometric construction and properties of some families of solutions of nonlinear partial differential equations (I), J. Math. Phys., 24 (1983), pp. 510–521. [7] F. JOHN,Partial Differential Equations, Springer-Verlag, New York, 1975. [8] T. A. SNEDDON,Elements of Partial Differential Equations, McGraw–Hill, New York, 1957. [9] F. TRICOMI,Equazioni a derivate parziali, Cremonese, Rome, Italy, 1957. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php