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Nowadays the Euler-Poisson-Darboux equation is extensively studied in severalsettings. The main questions on avery spaces are explicit solutions for theclassical Cauchy problems with the second data null. In this note we willgeneralize and unify several results on Euler-Poisson-Darboux equation. In thisnote we consider the Cauchy problem with modified conditions for theEuler-Poisson-Darboux equation. We give the explicit solutions in terms ofGauss 2F1 and Appel F4 hypergeometric functions with applications to theclassical and radial wave equations

Probl\`eme de Cauchy avec des conditions modifi\'ees pour les \'equations d'Euler-Poisson-Darboux