The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt (?-1784). According to I. Grattan-Guinness et S. Engelsman, The manuscripts of Paul Charpit, Historia Mathematica, vol. 9, 1982, p. 65-75:
He presented his paper on partial differential equations to the
Académie on June 30, 1784, but he died in Paris on December 28 of that
The memoir was lost, then found again. Lagrange learnt about it in 1793 only. There is a series of papers on that topic by N. Saltykow, such as Méthodes classiques d'intégration des équations aux dérivées partielles du premier ordre à une fonction inconnue, 1931. Page 7:
Charpit a eu, d'abord, la chance de formuler, le premier, les
équations différentielles ordinaires des caractéristiques, que l'on
attribue fréquemment à Lagrange
with the poor translation:
Charpit was lucky enough to be the first to express the ordinary differential
equations of characteristics, which are often attributed to Lagrange.
Full proof (in a modern sense) was apparently given by Jacobi.