Partial differential analogs of ordinary differential equations and systems
The authors deal with the completely integrable overdetermined systems \[ \partial x/\partial x^i= f^i(x,z)\quad (x\in\mathbb{R}^n,\;z\in\mathbb{R}^m), \] where the solutions are ensured by the Frobenius theorem and then with the analogous higher-order generalizations with a single unknown function \((z\in\mathbb{R})\). The method of variation of constants, Wronskians, Liouville formula, fundamental set of solutions are introduced for the particular linear subcase. Explicit solutions for linear systems with constant coefficients are obtained. In other words, the authors simulate the well-known results for the ordinary differential equations \[ dz/dx= f(x,z),\quad x\in\mathbb{R},\quad z\in\mathbb{R}^m, \] or \[ d^nz/dx^n= f(x,z,\dots, d^{n-1} z/dx^{n-1}),\quad x,z\in \mathbb{R}, \] for the case of several independent variables.
scientific article; zbMATH DE number 2166316