Operator factorization and the solution of second-order linear ordinary differential equations

References

• Boas , ML . 1983 . Mathematical Methods in the Physical Sciences , New York : Wiley .
   Google Scholar
• Margenau , H and Murphy , GM . 1956 . The Mathematics of Physics and Chemistry, , 2nd , New Jersey : Van Nostrand .
   Google Scholar
• Powles , JG . 1990 . Teaching the prototype differential equation of physics . European Journal of Physics , 11 : 323
   Google Scholar
• Jeffrey , A . 1996 . Mathematics for Engineers and Scientists, , 5th , London : Chapman and Hall .
   Google Scholar
• Hildebrand , FB . 1962 . Advanced Calculus for Applications , New Jersey : Prentice-Hall .
   Google Scholar
• Ayres , F . 1952 . Schaum's Outline of Theory and Problems of Differential Equations , New York, London : Schaum .
   Google Scholar
• Polya , G . 1922 . On the mean-value theorem corresponding to a given linear homogeneous differential equation . Transactions of the American Mathematical Society , 24 : 312
   Google Scholar
• Mammana , G . 1926 . Sopra un nuovo metodo di studio delle equazioni differenziali lineari . Math. Zeit , 25 : 734 1931, Decomposizione delle espressioni differenziali lineari omogenee in prodotti di fattori simbolici e applicazione relativa allo studio delle equazioni differenziali lineari, ibid, 33, 186
   Google Scholar
• Schrödinger , E . 1940 . A method of determining quantum mechanical eigenvalues and eigenfunctions . Proceedings of the Royal Society of Irish Acad , 46A : 9 Further studies on solving eigenvalue problems by factorization, ibid, 46 A, 183; 1941, The factorization of the hypergeometric equation, ibid, 47 A, 53
   Google Scholar
• Infeld , L and Hull , TE . 1951 . The factorization method . Reviews of Modern Physics , 23 : 21
   Web of Science ®Google Scholar
• Dita , P and Grama , N . 1997 . On Adomian's decomposition method for solving differential equations . arXiv:solv-int/9705008 v1 , 14 May
   Google Scholar
• Cayley , A . 1886 . On linear differential equations (the theory of decomposition) . Quarterly Journal of Mathematics , 21 : 331
   Google Scholar
• Davis , HT . 1962 . Introduction to Nonlinear Differential and Integral Equations , New York : Dover .
   Google Scholar
• Ince , EL . 1956 . Ordinary Differential Equations , New York : Dover .
   Google Scholar
• Kamke , E . 1944 . Differentialgleichungen: Lösungsmethoden und Lösungen , Leipzig : Akademische Verlagsgesellschaft .
   Google Scholar
• Murphy , GM . 1960 . Ordinary Differential Equations and their Solutions , New Jersey : Van Nostrand .
   Google Scholar
• Carinena , JF and Ramos , A . 1999 . Integrability of the Riccati equation from a group-theoretical viewpoint . Int. J. Mod. Phys. , A14 : 1935
   Web of Science ®Google Scholar
• Hong-Xiang , Li. 1982 . Elementary quadratures of ordinary differential equations . The American Mathematical Monthly , 89 : 198
   Web of Science ®Google Scholar
• Hoyt , JP . 1967 . Remarks on a second order differential equation . The American Mathematical Monthly , 74 : 436
   Web of Science ®Google Scholar
• Lynch , MAM and Gao , H . 2000 . Setting up second-order variable coefficient differential equation problems with known general solution . International Journal of Mathematical Education in Science and Technology , 31 : 727
   Google Scholar
• Archer-Hall , JA . 1970 . A class of second order non-linear differential equations with analytical solution . Mathematical Gazette , 54 : 60
   Google Scholar
• Underwood , F . 1932 . A class of homogeneous linear differential equations of the second order . Mathematical Gazette , 16 : 122
   Google Scholar
• Aly , HH and Barut , AO . 1990 . Methods of solution of the Schrödinger equations for a class of anharmonic and singular potentials . Physics Letters A , 145 : 299
   Web of Science ®Google Scholar
• Byrd , JW . 1976 . Introducing Green's function for initial- and boundary-value problems . American Journal of Physics , 44 : 596
   Web of Science ®Google Scholar
• Rainville , ED . 1972 . Intermediate Differential Equations , New York : MacMillan .
   Google Scholar
• Deeba , E , Yoon , J-M and Zafris , V . 2003 . An iterative method for solving variable coefficient ODEs . International Journal of Mathematical Education in Science and Technology , 34 : 908
   Google Scholar
• Forsyth , AR . 1997 . A Treatise on Differential Equations , New York : Dover .
   Google Scholar
• Mathews , J and Walker , RL . 1970 . Mathematical Methods of Physics , New York : W.A. Benjamin .
   Google Scholar
• Hyouguchi , R , Seto , R , Ueda , M and Adachi , S . 2004 . Divergence-free WKB theory . Annals of Physics , 312 : 177
   Web of Science ®Google Scholar
• Sukhatme , U and Pagnamenta , A . 1991 . Supersymmetry-inspired WKB approximation in quantum mechanics . American Journal of Physics , 59 : 944
   Web of Science ®Google Scholar
• Aharonov , Y and Au , CKY . 1979 . New approach to perturbation theory . Physical Review Letters , 42 : 1582 1979, Logarithmic perturbation expansions, Physical Review A, 20, 2245
   Web of Science ®Google Scholar
• Dicke , RH and Wittke , JP . 1969 . Introduction to Quantum Mechanics , Reading, Massachusetts : Addison-Wesley .
   Google Scholar
• Pellicciaro , EJ . 1970 . A representation formula for the solutions of the second order linear differential equation . Mathematical Magazine , 43 : 77
   Google Scholar
• Carinena , JF and Ramos , A . 2000 . Riccati equation, factorization method and shape invariance . Reviews in Mathematical Physics , 12 : 1279
   Web of Science ®Google Scholar
• Haaheim , DR and Stein , FM . 1969 . Methods of solution of the Riccati differential equation . Mathematical Magazine , 42 : 233
   Google Scholar
• De Oliveira , EC and Ducati , GC . 1999 . Calculation of Green's function for the Jacobi differential equation . International Journal of Mathematical Education in Science and Technology , 30 : 753
   Google Scholar
• Sawyer , WW . 1951 . The remainder theorem for operators . Mathematical Gazette , 35 : 47
   Google Scholar
• Sandell , DC and Stein , FM . 1977 . Factorization of operators of second order linear homogeneous ordinary differential equations . Two-Year College Mathematics Journal , 8 : 132
   Google Scholar
• Heinbockel , JH . 1964 . Factorization of linear second order differential operators . Mathematical Magazine , 37 : 302
   Google Scholar
• Jafarizadeh , MA and Fakhiri , H . 1997 . Supersymmetry and shape invariance in differential equations of mathematical physics . Physics Letters A , 230 : 164
   Web of Science ®Google Scholar
• Dutt , R , Khare , A and Sukhatme , UP . 1988 . Supersymmetry, shape invariance, and exactly solvable potentials . American Journal of Physics , 56 : 163
   Web of Science ®Google Scholar
• Simmons , GF . 1988 . Differential Equations , New Delhi : TATA McGraw-Hill .
   Google Scholar
• Ramankutty , P . 1991 . The complementary function and the general solution . Mathematical Magazine , 64 : 124
   Google Scholar
• Chandramohan , D and Balasubramanian , S . 1983 . On the method of solving second order linear differential equations . European Journal of Physics , 4 : 90
   Google Scholar
• Mukherjea , A . 1988 . A note on linear homogeneous differential equations . International Journal of Mathematical Education in Science and Technology , 19 : 625
   Google Scholar
• Humi , M . 1968 . Extension of the factorization method . The Mathematics of Physics , 9 : 1258 1970, New types of factorizable equations, Proceedings of the Cambridge Philosophical Society, 68, 439; 1986, Factorization of systems of differential equations, The Mathematics of Physics, 27, 76; 1987, Novel types of factorizable systems of differential equations, Journal of Physics A, 20, 1323
   Web of Science ®Google Scholar
• Piaggio , HTH . 1956 . Differential Equations , London : Bell .
   Google Scholar
• Ward , AJB . 1984 . Solution by quadratures of certain ordinary differential equations . International Journal of Mathematical Education in Science and Technology , 15 : 61
   Google Scholar
• Haley , SB . 1997 . American Journal of Physics , 65 : 237
   Web of Science ®Google Scholar
• Bose , AK . 1963 . Solvable potentials . Physics Letters , 7 : 245
   Google Scholar
• Lima , ML and Mignaco , JA . 1984 . Journal of Physics A , 17 : L323
   Google Scholar