By R. Courant and D. Hilbert
Chapter 1 Introductory comments (pages 1–61):
Chapter 2 normal thought of Partial Differential Equations of First Order (pages 62–131):
Chapter three Diflerential Equations of upper Order (pages 154–239):
Chapter four strength conception and Elliptic Differential Equations (pages 240–406):
Chapter five Hyperbolic Differential Equations in self sufficient Variables (pages 407–550):
Chapter 6 Hyperbolic Differential Equations in additional Than self reliant Variables (pages 551–798):
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Additional resources for Methods of Mathematical Physics: Partial Differential Equations, Volume II
If, however, we start with a two-parameter family of funct,ions 24, = f(x, 3 ; 0) ff, rather than with a set depending on an arbitrary function, and if we form the derivatives uz = f&, Y; a,P ) , uu = f&, 9 ; a,P ) , then we have three equations from which we usually can eliminate and @ (certainly if fzafva - fz,+,fUa # 0). We obtain a partial differential equation F(x,y, u , uz,u,) = 0 which, in general, is no longer linear in uz and uu . The paradox 'that a more restricted class of prescribed solutions leads to a more general type of differential equations will be resolved in $4.
U,, = 0 = u,); D i t follows that, for an integral siirface, either = Urr71,yy - U,2 =0 1 Incidentally, the differential equation (12) can be reduced by the substitution z = 1*/2 to the form (9) and thus solved; this may also be done hy means of the complete integral 71 = 03: + n1 y + L. - 38 I. INTRODUCTORY REMARKS or y = -f*. z = -fp, But the latter possibility yields precisely the exceptional surface obtained by the Legendre transformation. As another example, we consider the second order differential equation of minimal surfaces (see also Vol.
1 - - where D is the determinant of the symbols P i , Qi, - , and G' is a suitable symbolic linear combination of the functions gy. ). The symbols Gj likewise denote differential operators corresponding to the minors of the determinant D. , if the polynomials Pi , Qi , - - are linear, then the resulting equations are, in general, of order n. Suppose u is a solution of one of the resulting higher order equations; then after substituting u in the given system, we may omit one of the original equations, since the system now is dependent.