Geodesic
From differential equations to difference equations
Matematičeskoe obrazovanie, Tome 107 (2023) no. 3, pp. 38-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

The most popular, widely used method in regards to solving first order linear equations is through induction. However there are similar techniques that can be employed to obtain a general solution without the use of mathematical induction. Also, general solutions can be provided by borrowing a method based on the characteristic equation for second order linear difference equations of the Euler-Cauchy type. These differential equations are an important aspect of learning as they provide a fundamental foundation of tools and intuition that lead to partial differential equations which are used to describe phenomena in natural sciences. 
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U. Goginava; F. M. Mukhamedov. From differential equations to difference equations. Matematičeskoe obrazovanie, Tome 107 (2023) no. 3, pp. 38-47. http://geodesic.mathdoc.fr/item/MO_2023_107_3_a6/

[1] C. R. Adams, “On the irregular cases of linear ordinary difference equations”, Trans. Amer. Math. Soc., 30 (1928), 507–541 <ext-link ext-link-type='doi' href='https://doi.org/10.1090/S0002-9947-1928-1501443-6'>10.1090/S0002-9947-1928-1501443-6</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1501443'>1501443</ext-link>

[2] G. D. Birkhoff, “Formal theory of irregular linear drfference equations”, Acta Math., 54 (1930), 205–246 <ext-link ext-link-type='doi' href='https://doi.org/10.1007/BF02547522'>10.1007/BF02547522</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1555307'>1555307</ext-link>

[3] S. Elaydi, An Introduction to Difference Equations, Springer Science+Business Media, Inc., 2005 <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=2128146'>2128146</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:1071.39001'>1071.39001</ext-link>

[4] S. Goldberg, An Inroduction to Difference Equations, Wiley, New York, 1958 <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=94249'>94249</ext-link>

[5] A. Hongyo, N. Yamaoka, “General solution of second-order linear difference equations of Euler type”, Opuscula Math., 37 (2012), 389–402 <ext-link ext-link-type='doi' href='https://doi.org/10.7494/OpMath.2017.37.3.389'>10.7494/OpMath.2017.37.3.389</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=3605618'>3605618</ext-link>

[6] G. Strang, “Sums and differences vs. integrals and derivatives”, College Math. J., 21 (1990), 20–27 <ext-link ext-link-type='doi' href='https://doi.org/10.1080/07468342.1990.11973278'>10.1080/07468342.1990.11973278</ext-link>

[7] R. Wong, H. Li, “Asymptotic expansions for second-order linear difference equations”, J. Comput. Appl. Math., 41 (1992), 65–94 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/0377-0427(92)90239-T'>10.1016/0377-0427(92)90239-T</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1195637'>1195637</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:0758.39005'>0758.39005</ext-link>

[8] A. Zygmund, Trigonemetric Series, Cambridge Univ. Press, 2002 <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=178296'>178296</ext-link>