Demidov, S. S.; Petrova, S. S.; Simonov, N. I. Ordinary differential equations. (Russian) Zbl 0719.34004 Mathematics of the XIXth century, Chebyshev’s direction in function theory, Ordinary differential equations, Variational calculus, Theory of finite differences, Moskva 1987, 80-183 (1987). [For the entire collection see Zbl 0616.00002.] Contents: 1. Progress in the development of the theory of ordinary differential equations in the 18th century; 2. The uniqueness and existence problem; 2.1. Cauchy’s work; 2.2. Development of the majorant method; 2.3. The Cauchy-Lipschitz method; 2.4. The method of successive approximations; 3. Integration of equations in quadratures; 3.1. Liouville and the Riccati equations; 3.2. New classes of integrable equations; 3.3. Sophus Lie and the problem of integrability of differential equations in quadratures; 3.4. Special solutions; 4. Linear differential equations; 4.1. General theory; 4.2. Boundary value problems. Sturm-Liouville theory. 4.3. The solution of equations in the form of series, and special functions; 5. The analytic theory of differential equations; 5.1. The origins of Cauchy theory. The works of Briot and Bouquet; 5.2. B. Riemann; 5.3. L. Fuchs; 5.4. H. Poincaré; 5.5. Nonlinear equations; 5.6. The results of Russian mathematicians; 5.7. P. Painlevé; 6. The qualitative theory of differential equations; 6.1. Poincaré’s qualitative theory; 6.2. Lyapunov’s theory of stability; 6.3. Further development of the qualitative theory of differential equations; Conclusion. MSC: 34-03 History of ordinary differential equations 01A55 History of mathematics in the 19th century Citations:Zbl 0616.00002 PDFBibTeX XML