Some functional differential equations

Sufficient conditions are given for the solution of the functional differential equations with associated boundary conditions \[ d y / d x = ∑ n = 0 ∞ a n y ( μ n x ) , y ( 0 ) = 1 , d y / d x = ∫ 0 ∞ a ( u ) y ( μ u x ) d u , y ( 0 ) = 1 dy/dx = \sum \limits _{n = 0}^\infty {{a_n}y\left ( {{\mu ^n}x} \right ),\qquad y\left ( 0 \right ) = 1, \\ dy/dx = \int _0^\infty {a\left ( u \right )y\left ( {{\mu ^u}x} \right )du},\qquad y\left ( 0 \right ) = 1} \] A discussion is also given of some possible solutions to the differential equations which do not satisfy the boundary conditions.