This volume, the fifth out of nine, continues the translation of "Treatise on Analysis" by the French author and mathematician, Jean Dieudonne. The author shows how, for a voluntary restricted class of linear partial differential equations, the use of Lax/Maslov operators and pseudodifferential operators, combined with the spectral theory of operators in Hilbert spaces, leads to solutions that are much more explicit than solutions arrived at through "a priori" inequalities, which are useless applications.