Differential equation

In Mathematical, a differential equation is a relation between one or more unknown functions and their Dérivée S. the order from a differential equation corresponds to the maximum degree of differentiation to which one of the unknown functions was subjected there.

The differential equations are used to build mathematical model of phenomena Physique S and biological, for example for the study of the Radioactivité or the Celestial mechanics . Consequently, the differential equations represent a vast field of study, as well in pure mathematics as in mathematics applied.

Definition

First examples

Even if it is not the discipline which gave birth to the differential equations, the Dynamique of the populations into famous in a simple way of the most accessible examples. Thus the study of an isolated population, in a producing medium of food in abundance, conduit with the model following for manpower X according to time T

$x\text{'}\; (T)\; =K\; \backslash ,\; X\; (T)\; \backslash ,$

I.e. the population growth x' (T) , is at every moment, proportional to the size of the population X (T) . The solutions of this equation reveal a phenomenon of Exponential growth. This example, with the simplest examples of differential equations are described in the article differential equation (elementary mathematics).

A more complex system, formed of two species, prey and predator, conduit with the equations of Lotka-Volterra

The manpower of the preys is X (T) , that of predatory the there (T) . One falls down on the preceding case if is null there. The quantity X (T)   there (T) is a probability of meeting, which influences a population negatively, positively on the other. At every moment, knowing the involved populations, one can describe the tendency. These two equations are coupled i.e. they should be solved together. Mathematically, they should be conceived like only one equation of unknown factor the couple (X (T), there (T)) . If the initial manpower of the populations is known, the later evolution is perfectly given. It is done along one of the curves of evolution illustrated opposite, which let appear a cyclic behavior.

One of the most famous differential equations is the fundamental relation of dynamics, of Newton: F = my , where m is the mass of a particle, F the force exerted on this one and has the acceleration which results from it. In the case of a rectilinear motion, if the undergone force is function of the position (for example in the case of a Ressort) one obtains an equation of the form

$\backslash \; displaystyle\; m\; \backslash \; frac\; \{d^2\; X\}\; \{dt^2\}\; =f\; (X)$

This time, to determine the movement perfectly, it is necessary to give oneself initial position and speed.

Differential equation, process of evolution and determinism

The characteristics of a system governed by a differential equation are the following ones:

• the possible states a priori for the system form a space of finished size, i.e. can be described by a finished number of variables. This space is the Espace of the phases. For example, to describe the movement of a particle in usual space, three variables are needed. For the movement of a solid, six are necessary.
• the laws which control the temporal evolution are derivable functions at least .
• the evolution of the system is deterministic : knowing the initial conditions , i.e. the state of the system at time present, one can deduce the state from it from the system at any time of the future or past.

The deterministic aspect of the differential equations has particularly strong implications, and is concretized mathematically by the Théorème of Cauchy-Lipschitz.

The ordinary differential equations (sometimes represented by initials EDO) must be distinguished from the partial derivative equations (EDP), where is there function of several variables and where intervene of the derivative partial. These last have a space of state of infinite size and are not necessarily any more of the deterministic processes of evolution.

General standard

That is to say E a vector Space of finished size.

By definition , a differential equation (sometimes: ordinary differential equation) is a equation following form

$F\; (X,\; there,\; y\text{'},\; \backslash \; dowries,\; y^\; \{(N)\})\; =0$

where F is a continuous function on open a U of $\backslash \; mathbb\; \{R\}\; \backslash \; times\; E^n$, called field.

The order of this differential equation is the order there appearing N higher derivative. Are there a definite function of X of an interval I in E and $y\text{'},\; there$, \ ldots, y^ {(N)}\, Derived S successive from the function there . This function is known as solution there if it is of class $\backslash \; mathcal\; C^n$ and if

$\backslash \; forall\; X\; \backslash \; in\; I,\; \backslash \; qquad\; F\; (X,\; there\; (X),\; y\text{'}\; (X),\; \backslash \; dowries,\; y^\; \{(N)\}(X))=0$

To solve a differential equation amounts finding the functions solutions there . For example, the differential equation there ''+ = 0 has a general solution of the form there: there (X) = has .cos X + B . sin X , where has, B are constants (which one can determine if one adds of the initial conditions).

In a differential equation, the function can there be with actual values, or values in a vector Space of finished size, thus if has there as components y₁ and y₂ :

y'_2&=xy_1-2y_2 \ end {boxes}

The use in physics is of then speaking about system of coupled differential equations . But the fertile point of view in mathematics is there to see only one equation, for a function with vectorial values.

One can still widen the definition, by considering differential equations on differential varieties.

Solutions

Lifespan

If is solution of a differential equation there on interval I, one can consider his restriction on an interval J included in I. This one will remain solution of the differential equation of course.

It is often judicious to consider only the maximum approaches , i.e. those which are not the restrictions of any other.

It should not be believed in so far as the maximum approaches are defined on $\backslash \; mathbb\; \{R\}$ whole. It is completely possible that they have a finished lifespan in the future or the past. It is thus solutions of the equation y'=y², for example.

However, if a solution remains confined in a compact field, then it has one infinite lifespan.

Differential equation in solved form

A differential equation of order N is put in solved form when one can express the strongest derivative according to X and of the derivative the preceding ones

$y^\; \{(N)\}=G\; (X,\; there,\; y\text{'},\; \backslash \; dowries,\; y^\; \{(n-1)\})$

where G is a continuous function.

Example

The scalar differential equation of order 1 pennies forms solved: there '= G (X, there) , admits a simple geometrical interpretation in the plan brought back to a reference mark of axes (OX) , (OY) . One can represent, attached in each point of coordinates X, there , the vector of components 1 and G (X, there) , which constitutes a field of vectors plan. The curves solutions are charts of functions there = F (X) , continuously derivable, whose tangent in each point is given by the field of vectors.

Form solved and forms implicit

The differential equations which can be put in solved form enjoy good theoretical properties, with a theorem of existence and unicity of solutions: the Theorem of Cauchy-Lipschitz.

In the contrary case one says that the differential equation is under implicit form . One tests, on the largest possible fields, to put the differential equation in solved form. Then one must carry out the connection of the solutions obtained. The treatment of the differential equations of this type will be evoked at the end of the article.

Initial conditions, theorem of Cauchy-Lipschitz

A initial condition for an equation of order N of unknown factor is there the data of a value x₀ and N vectors Y₀,…, Y_n-1 . The function solution satisfies these initial conditions there if

$y\; (x\_0)\; =Y\_0,\; \backslash \; qquad\; y\text{'}\; (x\_0)\; =Y\_1,\; \backslash \; qquad\; \backslash \; dowries\; \backslash \; qquad\; y^\; \{(n-1)\}(x\_0)\; =Y\_\; \{n-1\}$

A problem of Cauchy is the data of a differential equation with a set of initial conditions.

For a differential equation in solved form , with the help of an assumption of regularity rather not very demanding (character locally lipschitzien with X fixed, compared to the block of the other variables), the Théorème of Cauchy-Lipschitz states that, for each initial condition

• it exists a solution which satisfies it and which is defined on an interval of the form $]\; x\_0-\; \backslash \; alpha,\; x\_0+\; \backslash \; alpha\; *\; there\; exists\; a\; single\; maximum\; approach\; which\; satisfies\; it$

Another traditional problem is that of the boundary conditions , for which one prescribes the values of a function solution in several points, even values of the limits of a function solution at the boundaries of the field. Thus the problem

$\backslash \; begin\; \{boxes\}\; there$ +y=0 \ \ there (0) =y (2 \ pi) =0 \ end {boxes}

Such a problem (sometimes called problem of Dirichlet ) can not have any solution very well or on the contrary an infinity of functions solutions.

Explicit resolution

See also: Solution of differential equations per squaring

The explicit resolution of the differential equations, using the usual functions and of the operator of primitivation, is seldom possible. A having small number of equations of the particular forms can be brought back by changes of successive variables to the simplest equation of all: the equation $y\text{'}=f\; \backslash ,$ which is a simple primitivation.

Among the differential equations being able to be entirely solved are the scalar linear equations of order a, the equations with separate variables, the homogeneous equations, the equation of Bernoulli, the vectorial differential equations with constant coefficients.

Others can be entirely solved since a particular solution is known, thus the linear differential equation of order two, the Differential equation of Riccati.

Properties of continuity of the solutions

Continuity compared to the initial conditions and with the parameters

The data of the initial conditions x₀ , Y₀,…, Y_n-1 defines a single function solution, function of X which one can note S (x₀, Y₀,…, Y_n-1, X) . One defines a total function thus S which gives an account of the way in which the solutions vary with the initial conditions. Its field of existence is an open .

While being placed on the assumptions of the theorem of Cauchy-Lipschitz, the solutions depend continuously on the initial conditions, i.e. the function S is a continuous function of the whole of its variables.

If one makes continuously depend the system on a parameter $\backslash \; lambda$, there is also continuity of S compared to this parameter. Indeed to add a parameter can be brought back to modify the system. It is enough to add a component $\backslash \; Lambda$ with the sought function, and to ask him to check the equation $\backslash \; Lambda\; \text{'}=0$ and the initial condition $\backslash \; Lambda\; (x\_0)\; =\; \backslash \; lambda$.

Total properties

That is to say there a particular solution of the differential equation, with for initial conditions x₀ , Y₀,…, Y_n-1 . The property of continuity makes it possible to give the behavior of the solutions corresponding to close initial conditions.

• if one restricts the solution with a segment x_f container x₀ , the solutions of close initial conditions form a tube of solutions around the solution there .

More precisely, for all $\backslash \; varepsilon\; >0$, there exists $\backslash \; eta\; >0$ such as if Z is solution with conditions initales x₀ , Z₀,…, Z_n-1 and the Z_i $\backslash \; eta$-proches of the Y_i , then the solution Z is traced in a tubular Voisinage of there , of ray $\backslash \; varepsilon$.

Consequently, if one takes a continuation z_n such solutions, of which the initial conditions tighten towards those of there , the continuation z_n converges uniformly towards there .

• if one studies the solution on all his field of existence, such a property is not checked any more.

Butterfly effect, chaos

See also: Theory of chaos

The preceding properties of continuity are to be handled with precaution, since they do not bring quantified information. In practice, one observes in many systems a extreme sensitivity on the long run to small initial variations, phenomenon popularized by Edward Lorenz under the name of Effet butterfly. To return account in a satisfactory way of the evolution of a physical system over a very long time, it would be necessary to push measurements of initial conditions until a precision inenvisageable. Thus it would be necessary to very include in the calculation of the weather forecasting of long run to the beats of the wings of butterfly.

The systems governed by differential equations, although being in theory deterministic, can raise behaviors extremely complex and appearing disordered, chaotic. Henri Poincaré was the first to clear up this concept of deterministic chaos. Its ideas will be long in being begun again, but are used now as base with the theory of the dynamic Systèmes.

Classifications

Autonomous differential equation

See also: vector Field

An important particular case that where the variable X does not appear in the functional equation, then is qualified autonomous : thus the equation there '= F (there) .

The laws of physics apply in general to functions of time, and are presented in the form of autonomous differential equations, which expresses the invariance of these laws in time. Thus if an autonomous system returns to its position intiale at the end of an time interval T , he consequently knows a periodic evolution of period T .

The study of the autonomous equations is equivalent to that of the fields of vectors. For a first order equation, the solutions are a family of curves which are not cut (according to the Théorème of Cauchy-Lipschitz) and which fills space. They are tangent with the field of vectors in each point.

See also Theorem of Poincaré-Bendixson.

Linear differential equation

See also: linear differential equation

A differential equation is known as linear when the expression of the equation is linear (or more generally refines) relative with the block of variables $(there,\; y\text{'},\dots \; y^\; \{(N)\})$. A scalar linear differential equation of order N and unknown factor will be there thus of the form

$a\_0\; there\; +\; a\_1\; y\text{'}\; +\; a\_2\; there$ +… + a_n y^ {(N)}= a_ {n+1} \,

where $a\_0\; \backslash ,$, $a\_1\; \backslash ,$,… $a\_n\; \backslash ,$, $a\_\; \{n+1\}\; \backslash ,$ are numerical functions.

A vectorial linear differential equation of order N will have the same aspect, by replacing the $a\_i$ by linear applications (or often of the matrices) functions of X . Such an equation will be sometimes also called linear differential connection.

Characteristics of the linear differential equations in solved form

• the solutions have one infinite lifespan.

• one can superimpose (to make linear combinations) of solutions of linear differential equations
• when the equation is homogeneous ($a\_\; \{n+1\}$ = 0), its whole of solutions is a vector space of dimension N time the dimension of E .
• it is thus enough to exhiber a sufficient number of solutions independent of the homogeneous equation to solve it. One can test the independence of solutions using the Wronskien.
• the whole of the solutions of the homogeneous equation is a space closely connected: the general solution is formed of the sum of this particular solution with the general solution of the associated homogeneous linear equation.
• the Méthode of variation of the constants allows, once solved the homogeneous equation, to solve the equation supplements
• in the case of equations with constant coefficients, one has explicit formulas of resolution using Exponentielle S of matrices or endomorphisms, or by using the transformation of Laplace.

Holomorphic differential equation

See also: holomorphic differential equation

A holomorphic differential equation is the counterpart, for the complex variable, of an ordinary differential equation. The general theory is much more complex.

Local results

A holomorphic differential equation in solved form checks the analog of the theorem of Cauchy-Lipschitz: local existence and unicity of a function solution, itself holomorphic.

Moreover if the equation depends on parameters in a holomorphic way, the solution too. There is also holomorphic dependence in the initial conditions.

However there is no more in general connection in a single maximum approach.

Total results

One has problems even for the differential equation simplest: the calculation of Primitive S. For example the construction of a function such as the Logarithme complexes is not univocal. One can seek to build determinations of the function logarithm on open the largest possible ones: for example of the split plans. One can also build a primitive “along a way”. The phenomenon of Monodromie appears then: if the way makes a turn in the direct direction around the origin, the primitive is modified of a constant (2iΠ). To give an account of the situation, it is necessary to utilize the concepts of Revêtement, Branchpoint.

The functions powers are also solutions of differential equations simple and likely to present monodromy. Thus the equation $z\; \text{'}=\; -\; z^3$ does not admit any holomorphic solution, nor even Méromorphe on the whole level.

Linear case

The theory of the linear holomorphic differential equations in solved form is very similar to that of the equations for the real variable, as long as one remains on fields Simplement related S. If not it also gives place to problems of the type Branchpoint.

Numerical methods

The resolution of the differential equations by squaring (i.e. using the elementary operations and of the primitivation) is not possible that in a number of very restricted cases. For example, even the scalar linear differential equations of order two do not admit such a formula of general resolution. It is thus essential to have techniques of approximate resolution.

Method of Euler

See also: Method of Euler

This method, oldest and simplest, also has a theoretical interest since it makes it possible to prove a result of existence of solutions under assumptions weaker than the theorem of Cauchy-Lipschitz: it is the Théorème of Cauchy-Peano-Arzela.

One considers a differential equation of order 1 pennies forms solved y'=f (X, there), with the initial condition there (x₀) =y₀.

The principle is to approach the solution there on B by a function closely connected per pieces, by operating a discretization of the parameter: one poses

$x\_i\; =\; a+ih$ where $h=\; (Ba)\; /n$ is the not .

The function closely connected per pieces will thus join the points of coordinates (x_i, y_i), and it acts to propose an algorithm to build the y_i starting from y₀. On each interval x_i+1 one takes for slope of the segment refines that which the equation suggests: F (x_i, y_i).

Other methods

See also: numerical Solution of differential equation

Most traditional are the Méthodes of Runge-Kutta, the Méthode of Newmark, the Méthode of the finished differences or the Finite element method which is adapted for the E.D.P.

Differential equation in implicit form

Treatment of an example

That is to say the implicit differential equation

$(y\text{'})\; ^2\; +\; xy\text{'}\; -\; there\; =\; 0\; \backslash ,$

To study it one carries out régionnement plan: one distinguishes the values (X, there) for which the equation T²+xT there =0 admits 0,1 or 2 solutions. One obtains three areas U , V , W . The area V is the Parabole of equation $y=-\; \backslash \; frac14\; x^2$, the areas U and W is both Ouvert S which it delimits.

One starts by being interested in the solutions which are traced only on one of the three fields

1. In the area U , the equation does not admit no solution.
2. There is a very whole solution traced in V , it is the singular solution $y=-\; \backslash \; frac14\; x^2$ traced in green opposite.
3. In open the W , the equation can be put under one of the two solved forms

$y\text{'}=\; \backslash \; frac\; \{-\; x+\; \backslash \; sqrt\; \{x^2+4y\}\}\; \{2\}\; \backslash \; qquad\; y\text{'}=\; \backslash \; frac\; \{-\; X\; \backslash \; sqrt\; \{x^2+4y\}\}\; \{2\}$ Each one of these two equations checks the Théorème of Cauchy-Lipschitz. If one restricts oneself with open the W , there are thus exactly two solutions for each couple of initial solutions. They are traced in blue on the figure opposite. In this case they are right-hand sides besides, of equation

$y\; =\; Ax\; +\; A^2\; \backslash ,$

They are tangent with the parabola of equation $y=-\; \backslash \; frac14\; x^2$. More precisely, the solutions traced on W are these lines, stopped at the tangential point since one leaves W .

One can now make the study of the differential equation on the whole level. There then exist “hybrid” solutions formed by connecting way $\backslash \; mathcal\; C^1$ an arc of parabola (green) with the rectilinear solutions (blue). Thus the solution represented in red:

Such a connection can be done only in one point of V . The description of the whole of all the solutions would be done by discussing according to the initial condition x₀, y₀

1. initial Condition in U : no the solution
2. initial Condition in V : for the values of X higher than x₀ , the solution can be the whole parabola, or then one follows an arc of parabola then one forks on the tangent. In the same way for the values lower than x₀ .
3. initial Condition in W : there are initially two lines solutions, tangents with the parabola. Either they indefinitely are prolonged, or one leaves them for the parabola on the level of the tangential point. One continues then on the parabola, or one sets out again on a tangent a little further.

Generalization

See also: implicit differential equation

To generalize this study it is necessary to be placed in a space at three dimensions, noted coordinates (X, there, p) . With the differential equation is associated the Surface with equation F (X, there, p) =0 (the coordinate p makes it possible to represent y' ). The solutions are curves plotted on surface. The encountered difficulties come from what these curves are projected on the plan (X, there) . The application of Projection knows critical points at the points where the Gradient of F is “vertical”. These are the points which are projected in the green parabola.

Finally, the framework of study is the same one as that of the theory of the envelope S. the parabola, solution singular is here the envelope of the family of the right-hand sides, regular solutions.

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