Let L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n +1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (-1)nQ*Q or L = (-l)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,…, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which y; and yn+j are mutually conjugate if i≠j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pòlya. © 1979, Royal Society of Edinburgh. All rights reserved.