Everything about Ordinary Differential Equation totally explained
In
mathematics, an
ordinary differential equation (or
ODE) is a relation that contains functions of only one
independent variable, and one or more of its
derivatives with respect to that variable.
A simple example is
Newton's second law of motion, which leads to the differential equation
»
Theories of ODEs
Singular solutions
The theory of
singular solutions of ordinary and partial
differential equations was a subject of research from the time
of Leibniz, but only since the middle of the
nineteenth century did it
receive special attention. A valuable but little-known work on the
subject is that of Houtain (1854). Darboux (starting in 1873) was a
leader in the theory, and in the geometric interpretation of these
solutions he opened a field which was worked by various
writers, notably
Casorati and Cayley. To the latter is due (1872)
the theory of singular solutions of differential equations of the
first order as accepted circa 1900.
Reduction to quadratures
The primitive attempt in dealing with differential equations had in view a reduction to
quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the
th degree, so it was the hope of analysts to find a general method for integrating any differential equation.
Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless
complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the
independent variable or variables, and if so, what are the characteristic properties of this function.
Fuchsian theory
Two memoirs by
Fuchs (
Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and
Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a
non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked
the theory along lines parallel to those followed in his theory of
Abelian integrals. As the latter can be classified according to the
properties of the fundamental curve which remains unchanged under a
rational transformation, so Clebsch proposed to classify the
transcendent functions defined by the differential equations
according to the invariant properties of the corresponding surfaces
f = 0 under rational one-to-one transformations.
Lie's theory
From 1870
Lie's work put the theory of differential equations
on a more satisfactory foundation. He showed that the integration
theories of the older mathematicians can, by the introduction of what are now called
Lie groups, be referred to a common source; and that
ordinary differential equations which admit the same
infinitesimal transformations present comparable difficulties of integration. He
also emphasized the subject of
transformations of contact.
Sturm-Liouville theory
Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients.
Further Information
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