Solving of Partial Differential Equations under Minimal Conditions - Maslyuchenko, V. K.; Mykhaylyuk, V. V. (Журнал математичної фізики, аналізу, геометрії, 2008)
It is proved that a differentiable with respect to each variable function f: R^2→ R is a solution of the equation ux+uy=0 if and only if there exists a function φ: R→ R such that f(x,y)=φ(x-y). This gives a positive answer to a question by R. Baire. Besides, this result is used to solve analogous partial differential equations in abstract spaces and partial differential equations of higher-order. Mathematics Subject Classification: 26B05, 35A99.