III. On linear differential equations.— No. II

The principles laid down in my former paper will enable us to integrate a proposed differential equation, when the solution can be expressed in the form—P/Q, where P, Q, to are rational and entire functions of ( x ). Let (α 0 +α 1 x +α 2 x 2 + ... +α m x m ) d n y / dx n + (β 0 +β 1 +β 2 x 2 + ... +α m x m ) d n-1 y / dx n-1 + (γ 0 +γ 1 x +γ 2 x 2 + ... +γ m x m ) d n-2 y / dx n-2 +.... +(λ 0 +λ 1 +λ 2 x 2 + ... +λ m x m ) y =0 be the general linear differential equation of the nth order, where none of the indices of ( x ) in the coefficients of the succeeding terms are greater than those in the coefficients of the two first.