 # Mutually conjugate solutions of formally self-adjoint differential equations

Journal Article

### Abstract

• 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.

• 1979

### Citation

• Hill, J. M., & Nillsen, R. V. (1979). Mutually conjugate solutions of formally self-adjoint differential equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 83(1-2), 93-101. doi:10.1017/S0308210500011409

### Scopus Eid

• 2-s2.0-84975988614

• 93

• 101

• 83

• 1-2

### Abstract

• 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.

• 1979

### Citation

• Hill, J. M., & Nillsen, R. V. (1979). Mutually conjugate solutions of formally self-adjoint differential equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 83(1-2), 93-101. doi:10.1017/S0308210500011409

### Scopus Eid

• 2-s2.0-84975988614

• 93

• 101

• 83

• 1-2