, h In the XIX century, mathematical physics continued to be the main
source of new partial differential equations and ofproblems involving
them. The study ofLaplace's equation and ofthe wave equation had assumed
a more systematic nature. In the beginning of the century, Fourier added
the heat equation to the aforementioned two. Marvellous progress in
obtaining precise solution repre- sentation formulas is connected with
Poisson, who obtained formulas for the solution of the Dirichlet problem
in a disc, for the solution of the Cauchy problems for the heat
equation, and for the three-dimensional wave equation. The physical
setting ofthe problem led to the gradual replacement ofthe search for a
general solution by the study of boundary value problems, which arose
naturallyfrom the physics ofthe problem. Among these, theCauchy problem
was of utmost importance. Only in the context of first order equations,
the original quest for general integralsjustified itself. Here again the
first steps are connected with the names of D'Alembert and Euler; the
theory was being intensively 1h developed all through the XIX century,
and was brought to an astounding completeness through the efforts
ofHamilton, Jacobi, Frobenius, and E. Cartan. In terms of concrete
equations, the studies in general rarely concerned equa- tions of higher
than second order, and at most in three variables. Classification 'h
ofsecond orderequations was undertaken in the second halfofthe XIX
century (by Du Bois-Raymond). An increase in the number of variables was
not sanc- tioned by applications, and led to the little understood
ultra-hyperbolic case.