Euler–Poisson–Darboux equation

In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation

${\displaystyle u_{x,y}+{\frac {N(u_{x}+u_{y})}{x+y}}=0.}$

This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave equation.

This equation is related to

${\displaystyle u_{rr}+{\frac {m}{r}}u_{r}-u_{tt}=0,}$

by ${\displaystyle x=r+t}$, ${\displaystyle y=r-t}$, where ${\displaystyle N={\frac {m}{2}}}$ and some sources quote this equation when referring to the Euler–Poisson–Darboux equation.