### Ordinary Differential Equations

L. S. Pontryagin, Leonas Kacinskas, Walter B. Counts, Dagmar Renate Henney
1963 Physics today
When studying ordinary differential equations, we should aim at a higher level, partial differential equations, so that our approach to the study is directed toward that goal. We may discover that there is one principle which runs through the entire theory of differential equations if we have the right attitude toward solving them. In order to solve differential equations, we may want to classify differential equations by forms at first. In other words, we would like to create a table of
more » ... e a table of solutions for differential equations. The more entries the table contains, the better it seems to be. In fact, this is not always true. For Bessel's equation alone, it has countless variations. No advantage is to be gained by studying these variations rather than Bessel's equation. Consequently, rather than by superficial forms, we should classify differential equations by essence: ordinary points, regular singularities, and irregular singularities. The rest of all the classifications and theories of differential equations can be considered supplements or generalizations of this view. We will discuss specific methods of solving ODEs, linear ODEs vs. linear systems (homogeneous case, Wronskians, reduction, adjoints, Green's second identity, nonhomogeneous case), the existence and uniqueness of solutions of ODEs, maximal intervals of existence, maximal or minimal solutions, sequences of ODEs, asymptotic behavior of solutions of an ODE, stability from the viewpoint of effectiveness, singularities, indicial equations, the Fuchsian type, self-adjoint eigenvalue problems, comparison theorems, Sturm's oscillation theorems, Green's functions, characteristic numbers for the Sturmian system, characteristic numbers for periodic boundary conditions where the coefficients of the system's equation are periodic, stable regions, self-adjoint boundary-value problems for secondorder singular equations, limit-point type, limit-circle type, the expansion and completeness theorem, Riccati's equation, dualty of adjoint systems, solving high order differential equations with Lommel's formula, finding the cases that Riccati's equation is integrable in finite terms, studying Sturm-Liouville Problems with the effective Ritz method, and miscellaneous notes. Keywords. Specific effective methods, homogeneous, the Euler linear equation, differentiation under the integral sign, Liouville's theorem, Wronskians, variation of constants, the method of successive approximation, reduction of order, reduction to smaller systems, adjoint equations, adjoint systems, Green's second identity, nonhomogeneous, existence and uniqueness, maximal interval of existence, maximal or minimal solutions, initial value problems, Lipschitz conditions, generalized Lipschitz conditions, limits, asymptotic behavior of solutions, stability, asymptotic stability, Lyapunov's theorem, ordinary point, simple singularity, singular points of the first (second) kind, regular singularities, indicial equations, the Fuchsian type, selfadjoint eigenvalue problems, comparison theorems, the Picone formula, Sturm-Liouville equation, Sturm's oscillation theorems, Green's functions, the Prüfer substitution, characteristic numbers for the Sturmian system, characteristic numbers for periodic boundary conditions, stable regions, when the coefficients of the system's equation are periodic, Ascoli's lemma, the Jordan canonical form, the expansion and completeness theorem, the Leibniz integral rule, entire function, Taylor series, interior point, the Pragmen-Lindelöf