General exact solutions of the second-order homogeneous algebraic differential equations

Abstract

The second-order homogeneous algebraic differential equation is studied, where are known such that is non-singular for some . A formula for the general exact solutions of the above equation is provided under the condition that there exists , such that . For a special second-order homogeneous algebraic differential equation coming from a physics model investigated by Bhat and Bernstein, a detailed description is given about how to construct a particular solution of the latter quadratic matrix equation.

Keywords:

• Drazin inverse
• quadratic matrix equation
• second-order algebraic differential equation

AMS Subject Classifications:

• 15A09
• 15A24
• 65L80

Acknowledgements

The authors thank Professor Nung-Sing Sze for his help. They also thank the referee for many detailed suggestions and helpful comments, which have improved this paper significantly.

Notes

1   Note that by (4545 ) any solution of is nonzero.

Supported by the National Natural Science Foundation of China [grant number 11171222] and the Innovation Program of Shanghai Municipal Education Commission [grant number 12ZZ129].