Ordinary differential equation
In mathematics, an ordinary differential equation (or ODE ) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable.
A simple example is Newton's cond law of motion, which leads to the differential equation
for the motion of a particle of constant mass m . In general, the force F depends upon the position x(t) of the particle at time t , and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F (x (t )).
Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of functions of veral variables.
Ordinary differential equations ari in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations
and contributed to the field,including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.
Much study has been devoted to the solution of ordinary differential equations. In the ca where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (e numerical ordinary differential equations).
The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from
Newton's cond law.
Existence and uniqueness of solutions
There are veral theorems that establish existence and
uniqueness of solutions to initial value problems
involving ODEs both locally and globally. See
Picard –Lindelöf theorem for a brief discussion of this
issue.
Definitions
Ordinary differential equation
Let y
be an unknown function in x with the n th derivative of y , and let F
be a given function
then an equation of the form
is called an ordinary differential equation (ODE) of order n . If y is an unknown vector valued function
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,
it is called a system of ordinary differential equations of dimension m (in this ca, F : ℝmn +1→ ℝm ).
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More generally, an implicit ordinary differential equation of order n
has the form
入党动机范文where F : ℝn+2→ ℝ depends on y(n). To distinguish the above ca from this one, an equation of the form
is called an explicit differential equation.
A differential equation not depending on x is called autonomous.
A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y together with a constant term, all possibly depending on x:
(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear with a
i
differential equation is called homogeneous, otherwi it is called non-homogeneous or inhomogeneous. Solutions
Given a differential equation
a function u: I⊂ R→ R is called the solution or integral curve for F, if u is n-times differentiable on I, and
Given two solutions u: J⊂ R→ R and v: I⊂ R→ R, u is called an extension of v if I⊂ J and
A solution which has no extension is called a global solution.
A general solution of an n-th order equation is a solution containing n arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by tting the constants to particular values, often chon to fulfill t 'initial conditions or boundary conditions'. A singular solution is a solution that can't be derived from the general solution.
Reduction to a first order system
Any differential equation of order n can be written as a system of n first-order differential equations. Given an explicit ordinary differential equation of order n (and dimension 1),
define a new family of unknown functions
for i from 1 to n.
The original differential equation can be rewritten as the system of differential equations with order 1 and dimension n given by
which can be written concily in vector notation as
with
and
Linear ordinary differential equations
A well understood particular class of differential equations is linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1
which we can write concily using matrix and vector notation as
with
Homogeneous equations
微心愿活动The t of solutions for a system of homogeneous linear differential equations of order 1 and dimension n
forms an n-dimensional vector space. Given a basis for this vector space , which is called a fundamental system, every solution can be written as
The n × n matrix
is called fundamental matrix. In general there is no method to explicitly construct a fundamental system, but if one solution is known d'Alembert reduction can be ud to reduce the dimension of the differential equation by one.冬天好玩的地方
Nonhomogeneous equations
The t of solutions for a system of inhomogeneous linear differential equations of order 1 and dimension n
can be constructed by finding the fundamental system to the corresponding homogeneous equation and one particular solution to the inhomogeneous equation. Every solution to nonhomogeneous equation can then be written as
A particular solution to the nonhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters.
Concerning cond order linear ordinary differential equations, it is well known that
So, if is a solution of: , then such that:
So, if is a solution of: ; then a particular solution of , is
given by:
. [1]
Fundamental systems for homogeneous equations with constant coefficients
If a system of homogeneous linear differential equations has constant coefficients
then we can explicitly construct a fundamental system. The fundamental system can be written as a matrix differential equation
with solution as a matrix exponential
which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal form
and then evaluate the Jordan blocks
of J parately as
Theories of ODEs怎么染头发
Singular solutions
杜仲猪腰The theory of singular solutions of ordinary and partial differential equations was a subject of rearch 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 the 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 th
e 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, subquently 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 tho 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 propod 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.
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Lie's theory
From 1870 Sophus 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 prent comparable difficulties of integration. He also emphasized the subject of transformations of contact.
A general approach to solve DE's us the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are ud to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the DE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
Sturm–Liouville theory
Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear operators defined in terms of cond-order homogeneous linear equations, and is uful in the analysis of certain partial
differential equations.
Software for ODE solving
•FuncDesigner (free licen: BSD, us Automatic differentiation, also can be ud online via Sage-rver [2])•VisSim [3] - a visual language for differential equation solving
•Mathematical Assistant on Web [4] online solving first order (linear and with parated variables) and cond order linear differential equations (with constant coefficients), including intermediate steps in the solution.•DotNumerics: Ordinary Differential Equations for C# and VB [5] Initial-value problem for nonstiff and stiff ordinary differential equations (explicit Runge-Kutta, implicit Runge-Kutta, Gear’s BDF and Adams-Moulton).•Online experiments with JSXGraph [6]
