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The linear symmetries of a nonlinear differential equation. (English) Zbl 0618.34009
Authors’ summary: ”The Lie point symmetries of the nonlinear differential equation \(\ddot q+3q\dot q+q^ 3=0\) which arises in the study of the modified Emden equation are shown to have an unexpectedly rich algebra associated with them, a feature which enables the above nonlinear differential equation to be transformed into the simplest linear second order ordinary differential equation.”
Reviewer: N.L.Maria

MSC:
34A34 Nonlinear ordinary differential equations and systems, general theory
22E70 Applications of Lie groups to the sciences; explicit representations
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[1] Anderson R. L., J. Math. Anal. Appl. 48 pp 301– (1974) · Zbl 0291.34007 · doi:10.1016/0022-247X(74)90236-4
[2] Axford R. A., Lectures on Lie Groups and Systems of Ordinary and Partial Differential Equations (1983)
[3] Bluman G. W., Similarity Methods for Differential Equations (1974) · Zbl 0292.35001 · doi:10.1007/978-1-4612-6394-4
[4] Djukic Dj. S., Acta Mech. 23 pp 17– (1975) · Zbl 0328.70013 · doi:10.1007/BF01177666
[5] Eisenhart L. P., Continuous Groups of Transformatations (1961)
[6] Fradkin D. M., Amer. J. Phys. 33 pp 207– (1965) · Zbl 0128.45601 · doi:10.1119/1.1971373
[7] Jauch J. M., Phys. Rev. 57 pp 641– (1940) · Zbl 0023.28503 · doi:10.1103/PhysRev.57.641
[8] Kobussen J. A., Helv. Phys. Acta. 53 pp 183– (1980)
[9] Leach P. G.L., Reflections on the symmetries and invariants of a one-dimensional linear system (1980)
[10] Leach P. G.L., J. Austral. Math. Soc. (B) 22 pp 12– (1980) · Zbl 0429.70017 · doi:10.1017/S0334270000002502
[11] Leach P. G.L., J. Austral. Math. Soc. (B) 23 pp 173– (1981) · Zbl 0479.70020 · doi:10.1017/S0334270000000151
[12] Leach P. G.L., J. Math. Phys. 22 pp 465– (1981) · Zbl 0459.70024 · doi:10.1063/1.524932
[13] Leach P. G.L., J. Math. Phys. 26 (1985)
[14] Leach P. G.L., J. Math. Phys. 26 (1985)
[15] Lenz W., Z. für. Phys. 24 pp 197– (1924) · doi:10.1007/BF01327245
[16] Leéy-Leblond J. M., Amer. J. Phys. 39 pp 502– (1971) · doi:10.1119/1.1986202
[17] Lewis H. R., Ann. Phys.
[18] Lie S., Vorlesungen über Differentialgleichungen (1967)
[19] Lovelock D., Tensors, Differential Forms and Variational Principles (1975) · Zbl 0308.53008
[20] Lutzky M., J. Phys. A. 11 pp 249– (1978) · Zbl 0369.22020 · doi:10.1088/0305-4470/11/2/005
[21] Moreira I. de C., Comments on ”A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians”
[22] Noether E., Nachr. kgl. Ges. Wiss. Göttingen Math. Phys. Kl. pp 235– (1918)
[23] Prince G. E., Hadronic J. 3 pp 941– (1980)
[24] Prince G. E., J. Phys. A. 13 pp 815– (1980) · Zbl 0432.70036 · doi:10.1088/0305-4470/13/3/015
[25] Prince G. E., J. Phys. A. 14 pp 587– (1981) · Zbl 0458.70001 · doi:10.1088/0305-4470/14/3/009
[26] Runge C. D.T., Vektoranalysis (1963) · JFM 47.0696.03
[27] Sarlet W., Sim Rev. 23 (4) pp 467– (1981) · Zbl 0474.70014 · doi:10.1137/1023098
[28] Wulfman C. E., J. Phys. A. 9 pp 507– (1976) · Zbl 0321.34034 · doi:10.1088/0305-4470/9/4/007
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