Solving of partial differential equations under minimal conditions

Abstract

It is proved that a differentiable with respect to each variable function $f:{\mathbb{R}}^2\to \mathbb{R}$ is a solution of the equation $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$ if and only if there exists a function $\phi :\mathbb{R}\to \mathbb{R}$ such that $f(x,y)=\phi (x-y)$. This gives a positive answer to a question of R.~Baire. Besides, we use this result to solving analogous partial differential equations in abstract spaces and partial differential equations of higher-order.