II. On the solution of linear differential equations

If the operation of differentiation with regard to the independent variable x be denoted by the symbol D, and if ϕ(D) represent any function of D composed of integral powers positive or negative, or both positive and negative, it may easily be shown, that ϕ(D){ψx. u} = ψx. ϕ(D)u + ψ'x. ϕ'(D)u + ½ψ"x. ϕ"(D)u + 1/2.3 ψ"'x. ϕ"'(D)u + . . . (1.) and that ϕx.ψ(D)u = ψ(D){ϕx. u} - ψ'(D){ϕ'x. u} + ½ψ"(D){ϕ"x. u} - 1/2.3ψ"'(D){ϕ"'x. u} + . . (2.) and these general theorems are expressions of the laws under which the operations of differentiation, direct and inverse, combine with those operations which are de­noted by factors, functions of the independent variable. It will be perceived that the right-hand side of each of these equations is a linear differential expression; and whenever an expression assumes or can be made to assume either of these forms, its solution is determined; for the equations ϕ(D){ψx. u} = P and ϕx. ψ(D)u = P are respectively equivalent to u = (ψx)-1 {ϕ(D)}-1 P and u = {ψ(D)}-1 ((ϕx)-1 P).