Compatible Systems and Charpit s Method

Size: px

Start display at page:

Download "Compatible Systems and Charpit s Method"

Garey Skinner
6 years ago
Views:

1 MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 28 Lecture 5 Compatible Systems Charpit s Method In this lecture, we shall study compatible systems of first-order PDEs the Charpit s method for solving nonlinear PDEs. Let s begin with the following definition. DEFINITION 1. (Compatible systems of first-order PDEs) A system of two firstorder PDEs f(x, y, z, p, q) 0 (1) g(x, y, z, p, q) 0 (2) are said to be compatible if they have a common solution. THEOREM 2. The equations f(x, y, z, p, q) 0 g(x, y, z, p, q) 0 are compatible on a domain D if (i) J (f,g) (p,q) (ii) g p 0 on D. p q can be explicitly solved from (1) (2) as p ϕ(x, y, z) q ψ(x, y, z). Further, the equation is integrable. ϕ(x, y, z) + ψ(x, y, z)dy THEOREM 3. A necessary sufficient condition for the integrability of the equation ϕ(x, y, z) + ψ(x, y, z)dy is g) + + p (f, + q 0. (3) (x, p) (y, q) (z, p) (z, q) In other words, the equations (1) (2) are compatible iff (3) holds. EXAMPLE 4. Show that the equations xp yq 0, z(xp + yq) 2xy are compatible solve them. Solution. Take f xp yq 0, g z(xp + yq) 2xy 0. Note that f x p, f y q, f z 0, x, y. g x zp 2y, g y zq 2x, g z xp + yq, g p zx, zy.

2 MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 29 Compute J (p, q) g p x zx y zy zxy + zxy 2zxy 0 for x 0, y 0, z 0. Further, f x (x, p) g x g p f z (z, p) g z g p f y (y, q) g y f z (z, q) g z p x zxp x(zp 2y) 2xy zp 2y zx 0 x xp + yq zx 0 x(xp + yq) x2 p xyq q y qzy + y(zq 2x) 2xy zq 2x zy 0 y xp + yq zy y(xp + yq) y2 q + xyp. It is an easy exercise to verify that g) + + p (f, + q (x, p) (y, q) (z, p) (z, q) 2xy x 2 p 2 xypq 2xy + y 2 q 2 + xypq y 2 q 2 x 2 p 2 0. So the equations are compatible. Next step to determine p q from the two equations xp yq 0, z(xp+yq) 2xy. Using these two equations, we have zxp + zyq 2xy 0 xp + yq 2xy z 2xp 2xy p y z z xp yq 0 q xp y xy yz x z q x z Substituting p q in p + qdy, we get ψ(x, y, z). z y + xdy d(xy), ϕ(x, y, z).

3 MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 30 hence integrating, we obtain z 2 2xy + k, where k is a constant. NOTE: For the compatibility of f(x, y, z, p, q) 0 g(x, y, z, p, q) 0 it is not necessary that every solution of f(x, y, z, p, q) 0 be a solution of g(x, y, z, p, q) 0 or vice-versa as is generally believed. For instance, the equations f xp yq x 0 (4) g x 2 p + q xz 0 (5) are compatible. They have common solutions z x + c(1 + xy), where c is an arbitrary constant. Note that z x(y + 1) is a solution of (4) but not of (5). Charpit s Method: It is a general method for finding the complete integral of a nonlinear PDE of first-order of the form f(x, y, z, p, q) 0. (6) Basic Idea: The basic idea of this method is to introduce another partial differential equation of the first order g(x, y, z, p, q, a) 0 (7) which contains an arbitrary constant a is such that (i) Equations (6) (7) can be solved for p q to obtain p p(x, y, z, a), q q(x, y, z, a). (ii) The equation p(x, y, z, a) + q(x, y, z, a)dy (8) is integrable. When such a function g is found, the solution F (x, y, z, a, b) 0 of (8) containing two arbitrary constants a, b will be the solution of (6). Note: Notice that another PDE g is introduced so that the equations f g are compatible then common solutions of f g are determined in the Charpit s method. The equations (6) (7) are compatible if g) + + p (f, + q (x, p) (y, q) (z, p) (z, q) 0.

4 MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 31 Exping it, we are led to the linear PDE g x + f g q y + (p + q ) g z (f x + pf z ) g p (f y + qf z ) g q 0. (9) Now solve (9) to determine g by finding the integrals of the following auxiliary equations: dy p + q dp (f x + pf z ) (f y + qf z ) (10) These equations are known as Charpit s equations which are equivalent to the characteristics equations (10) of the previous Lecture 4. Once an integral g(x, y, z, p, q, a) of this kind has been found, the problem reduces to solving for p q, then integrating equation (8). REMARK For finding integrals, all of Charpit s equations (10) need not to be used. 2. p or q must occur in the solution obtained from (10). EXAMPLE 6. Find a complete integral of p 2 x + q 2 y z. (11) Solution. To find a complete integral, we proceed as follows. Step 1: (Computing f x, f y, f z,, ). Set f p 2 x + q 2 y z 0. Then f x p 2, f y q 2, f z 1, 2px, 2qy. p + q 2p 2 x + 2q 2 y, (f x + pf z ) p 2 + p, (f y + qf z ) q 2 + q. Step 2: (Writing Charpit s equations finding a solution g(x, y, z, p, q, a)). The Charpit s equations (or auxiliary) equations are: dy 2px dy 2qy dp p + q (f x + pf z ) 2(p 2 x + q 2 y) dp p 2 + p (f y + qf z ) q 2 + q From which it follows that p 2 + 2pxdp 2p 3 x + 2p 2 x 2p 3 x p2 + 2pxdp p 2 x q2 dy + 2qy q 2 y q 2 dy + 2qy 2q 3 y + 2q 2 y 2q 3 y

5 MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 32 On integrating, we obtain log(p 2 x) log(q 2 y) + log a p 2 x aq 2 y, (12) where a is an arbitrary constant. Step 3: (Solving for p q). Using (11) (12), we find that p 2 x + q 2 y z, p 2 x aq 2 y (aq 2 y) + q 2 y z q 2 y(1 + a) z q 2 z (1 + a)y q z 1/2. (1 + a)y p 2 aq 2 y x a z (1 + a)y az 1/2 p. (1 + a)x y x az (1 + a)x Step 4: (Writing p(x, y, z, a) + q(x, y, z, a)dy finding its solution). Writing Integrate to have the complete integral of the equation (11). az 1/2 z 1/2 + dy (1 + a)x (1 + a)y ( ) 1 + a 1/2 ( a ) ( ) 1/2 1 1/2 + dy. z x y [(1 + a)z] 1/2 (ax) 1/2 + (y) 1/2 + b Practice Problems 1. Show that the equations xp yq x x 2 p + q xz are compatible solve them. 2. Show that the equations f(x, y, p, q) 0 g(x, y, p, q) 0 are compatible if 3. Find complete integrals of the equations: (i) p (z + qy) 2 ; (ii) (p 2 + q 2 )y qz + (x, p) (y, p) 0.

Compatible Systems and Charpit s Method Charpit s Method Some Special Types of First-Order PDEs. MA 201: Partial Differential Equations Lecture - 5

Compatible Systems and Charpit s Method Charpit s Method Some Special Types of First-Order PDEs. MA 201: Partial Differential Equations Lecture - 5 Compatible Systems and MA 201: Partial Differential Equations Lecture - 5 Compatible Systems and Definition (Compatible systems of first-order PDEs) A system of two first-order PDEs and f(x,y,u,p,q) 0