Ordinary differential equation

An ordinary differential equation ( ODE or ODE often abbreviated, English ordinary differential equation ) is a differential equation in which an unknown function only derivatives with respect to exactly one variable occur.

Many scientific models use ordinary differential equations in order to predict.

Origin

Differential equations are often required to describe processes in nature, in which the changing behavior of sizes is compared.

The first differential equations were the uniform and uniformly accelerated motion. In 1590 Galileo Galilei recognized the connection between the fall time of a body and its velocity of fall and the fall path and put it ( yet) with geometric means of the law of free fall.

When Isaac Newton also considered movements under the amount or square of the velocity -proportional friction, he was obliged to introduce the differential calculus and the now familiar formalism of differential equations.

The exact formulation of the limit concept, the derivative and the integral finally figured Augustin Louis Cauchy in the 19th century, the theory of ordinary differential equations on a firm foundation and thus opened it to many sciences.

Scientific interest in differential equations is essentially due to the fact that with them on the basis of relatively simple observations and experiments complete models can be created.

Only a few types of differential equations can be solved analytically. Nevertheless, qualitative statements such as stability, periodicity or bifurcation can also be taken if the differential equation can not be solved explicitly. One of the most important tools for scalar differential equations are arguments by means of a comparison set.

General definition

Let and be a continuous function. Then say

An ordinary differential equation system of order of equations. In case it is called an ordinary differential equation of order.

Their solutions are times differentiable functions which satisfy the differential equation on an interval to be determined. Addiction is a special solution to existing and additional

Met, this is referred to as an initial value problem.

The differential equation can be solved for the highest occurring discharge and thus has the form

It is called explicitly, otherwise implicitly; see also theorem of the implicit function.

Existence and uniqueness

Whether such a solution exists, it can be seen with reference to some criteria. The differential equation itself is not generally sufficient to determine the solution uniquely.

For example, the basic sequence of movements of all swinging pendulum is the same and can be described by a single equation. However, the exact sequence of movements is through the boundary or initial condition ( s) ( when the pendulum was initiated, and how large the initial displacement is ) determined.

The local solvability of initial value problems for ordinary differential equations of first order is described by the set of Picard - Lindelöf and the set of Peano. The existence of a local solution can be deduced in a second step, the existence of a non- continuable solution. With the help of the principle of maximum interval of existence can on this basis of this non- continuable solution then occasionally prove globality. The uniqueness one gets as an application of grönwallschen inequality.

Reduction of higher-order equations to systems of first order

Ordinary differential equations of arbitrary order always can be traced back to a system of ordinary differential equations of first order. Does an ordinary differential equation, the order, we will cause the following auxiliary functions

Thus one obtains a system of ordinary differential equations of first order. Conversely, one can derive a single differential equation of higher order of some differential equations.

Examples

• A simple example from physics is the decay law:

• An important class of other differential equations form the Newtonian equations of motion:

• In addition to simple contexts of the changes of a single size, but can also predict several sizes to meet in a system. In about the Lotka -Volterra equations of Ecology:

Special types of differential equations

The best-known type of ordinary differential equations is the linear differential equation of order with:

Another important type of ordinary differential equations include:

• D' Alembert equation

• Bernoulli differential equation

• Exact differential equation

• Jacobi differential equation

• Linear system of differential equations of first order equations

• Riccati differential equation

• Separable differential equation

Autonomous Systems

A differential equation system is autonomous, if of the form

Is.

Be linearly bounded and Lipschitz continuous. Denote the (uniquely determined global ) solution of

Then we call the river of the differential equation, and then forms a dynamic system.

Is the case of the planar autonomous systems of particular interest. With the help of the theorem of Poincaré - Bendixson you can often prove the existence of periodic solutions. An important plane autonomous system is the Lotka - Volterra model.

Since the Poincaré - Bendixson theory is based on the centrally between jordan curve theorem, higher-dimensional analogues are wrong. In particular, it is very difficult to find periodic solutions of higher dimensional autonomous systems.

Numerical Methods

As there are ordinary differential equations of higher order always reduce to systems of first order, you go out in the construction of numerical methods in the normal case of a first order system:

There are two important classes of numerical methods for the solution of ordinary differential equations, the one-step method (especially the Runge- Kutta method ) and linear multistep methods. A generalization of two classes represent the general linear procedure (general linear methods, abbreviated GLM ) dar.