This page presents a partial list of textbooks and papers that contain explanations of methods and algorithms for ODEs implemented in Maple's dsolve. The references on this page are listed in order of publication date within each section, with the most recent publication first.

For the methods based on classifying the ODE

Polyanin, A.D., and Zaitsev, V.F. Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, 1995.

Zwillinger, D. Handbook of Differential Equations. 2nd ed. Academic Press, 1992.

Murphy, G.M. Ordinary Differential Equations and their Solutions. Princeton: Van Nostrand, 1960.

Kamke, E. Differentialgleichungen: Loesungsmethoden und Loesungen. New York: Chelsea Publishing Co., 1959.

Ince, E.L. Ordinary Differential Equations. New York: Dover Publications, 1956.

For the Lie's symmetry method

Bluman, G.W. and Kumei, S., Symmetries and Differential Equations, Applied Mathematical Sciences New York: Springer-Verlag, 1989. Vol. 81.

Stephani, H. Differential Equations: Their Solution Using Symmetries. Edited by M.A.H. MacCallum.  New York and London: Cambridge University Press, 1989.

Olver, P.J. Applications of Lie Groups to Differential Equations. New York: Springer-Verlag, 1986.

Related to computational implementations for first order ODEs of Abel type

Cheb-Terrab, E.S. "A connection between Abel and pFq hypergeometric differential equations." European Journal of Applied Mathematics. Vol. 15. (2004): 1-11.

Cheb-Terrab, E.S., and Roche, A.D. "An Abel ordinary differential equation class generalizing known integrable classes." European Journal of Applied Mathematics. Vol. 14. (2003): 217-229.

Cheb-Terrab, E.S., and Roche, A.D. "Abel ODEs: Equivalence and  integrable classes." Computer Physics Communications. Vol. 130. (2000): 204-231.

Related to the computational implementation of integrating factor methods

Cheb-Terrab, E.S., and Roche, A.D. "Integrating Factors for Second Order ODEs." Journal of Symbolic Computation. Vol. 27 No. 5. (1999): 501-519.

Related to computational implementations for non-linear ODEs, including Lie symmetry methods

Avellar, J.; Cardoso, M. S., Duarte, L.G.S.; and da Mota, L.A.C.P. "Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The S-function method." Computer Physics Communications. Vol. 234. (2019): 302

Cheb-Terrab, E.S., and Kolokolnikov, T. "First-order ordinary differential equations, symmetries and linear transformations." European Journal of Applied Mathematics. Vol. 14. (2003):231-246.

Cheb-Terrab, E.S. "A Computational Approach for the Exact Solving of Systems of Partial Differential Equations." Submitted to Computer Physics Communications, 2001.

Cheb-Terrab, E.S., and Roche, A.D. "Symmetries and First Order ODE Patterns." Computer Physics Communications. Vol. 113. (1998): 239.

Cheb-Terrab, E.S.; Duarte, L.G.S.; and da Mota, L.A.C.P. "Computer Algebra Solving of Second Order ODEs Using Symmetry Methods." Computer Physics Communications. Vol. 108. (1998): 90.

Cheb-Terrab, E.S.; Duarte, L.G.S.; and da Mota, L.A.C.P. "Computer Algebra Solving of First Order ODEs Using Symmetry Methods." Computer Physics Communications. Vol. 101. (1997): 254.

Related to computational implementations of Special Function solutions for linear ODEs

Imamoglu, E. and van Hoeij, M. "Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases". Journal of Symbolic Computation, 83, (2017): 254-271.

Cheb-Terrab, E.S. and Roche, A.D., "Hypergeometric solutions for third order linear ODEs". Submitted for publication (2008).

Cheb-Terrab, E.S. "Solutions for the General, Confluent, and Bi-Confluent Heun equations and their connection with Abel equations." Journal of Physics A: Mathematical and General. Vol. 37. (2004): 9923-9949.

Chan, L., and Cheb-Terrab, E.S. "Non-Liouvillian solutions for second order linear ODEs." Proceedings of ISSAC'04, Santander, Spain. ACM Press. (2004): 80-86.

Burger, R.; Labahn, G.; and van Hoeij, M. "Closed form solutions of linear odes having elliptic functions as coefficients." Proceedings of ISSAC'04, Santander,  Spain. ACM Press. (2004): 58-64.

Chan, L., supervised by E.S. Cheb-Terrab. "On Solving second order linear ODEs admitting non-Liouvillian solutions." Report for NSERC University research award, Department of Mathematics, Simon Fraser University (2001).

von Bulow, K., supervised by E.S. Cheb-Terrab. "Equivalence Methods for Second Order Linear Differential Equations." Master's thesis, Faculty of Mathematics, University of Waterloo (2000).

Related to computational implementations for linear ODEs

Weil, J.A. "Recent Algorithms for Solving Second-Order Differential Equations." The Algorithm Project. (2002).

van Hoeij, M. "Factorization of Differential Operators with Rational Functions Coefficients." Journal of Symbolic Computation. Vol. 24. (1997): 537-561.

Labahn, G. "Solving Linear Differential Equations in Maple." MapleTech. Vol. 2 No. 1. (1995): 20.

Kamran, N., and Olver, P.J. "Equivalence of Differential Operators." SIAM J. Math. Anal. Vol. 20 No. 5. (1989): 1172.

Kovacic, J. "An algorithm for solving second order linear homogeneous equations." J. Symb. Comp. Vol. 2. (1986): 3-43.

Smith, Carolyn J., supervised by K.O. Geddes. "A Discussion and Implementation of Kovacic's Algorithm for Ordinary Differential Equations." Master's thesis, Faculty of Mathematics, University of Waterloo, 1984.

Related to computational implementations of elimination algorithms (singular solutions and triangularization of ODE and PDE systems)

Baechler, T., Gerdt, V., Lange-Hegermann, M. and Robertz, D., "Algorithmic Thomas decomposition of algebraic and differential systems." Journal of Symbolic Computation 47 (10) (2012): 1233-1266.

Reid, G.J.; Wittkopf, A.D.; and Boulton, A. "Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms." Eur. J. Appl. Math. (1996): 635.

Boulier, F.; Lazard, D.; Ollivier, F.; and Petitot, M. "Representation for the Radical of a Finitely Generated Differential Ideal." Proceedings of ISSAC '95. (1995): 158-166.