Obična diferencijalna jednačina

U matematici, obična diferencialna jednačina (ODE) je diferencijalna jednačina koja sadrži jednu ili više funkcija sa jednom nezavisnom promenljivom i izvodima tih funkcija. Termin obična se koristi u kontrastu sa terminom parcijalna diferencijalna jednačina, koja može biti definsana u odnosu na više od jedne nezavisne promenljive.

Diferencijalne jednačine

Linearna diferencijalna jednačina je diferencijalna jednačina koja je definisana pomoću linearnog polinoma u nepoznatoj funkciji i njegovih derivata. Drugim rečima to je jednačina oblika

$a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,$

gde su $a_{0}(x)$, ..., $a_{n}(x)$ i $b(x)$ proizvoljne fiferencijabilne funkcije koje ne moraju da budu linearne, a $y',\ldots ,y^{(n)}$ su sukcesivni derivati nepoznate funkcije y promenljive x.

Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation).

Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.

Zaleđina

The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law.

Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.

Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).

Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler.

A simple example is Newton's second law of motion — the relationship between the displacement x and the time t of an object under the force F, is given by the differential equation

$m{\frac {\mathrm {d} ^{2}x(t)}{\mathrm {d} t^{2}}}=F(x(t))\,$

which constrains the motion of a particle of constant mass m. In general, F is a function of the position x(t) of the particle at time t. The unknown function x(t) appears on both sides of the differential equation, and is indicated in the notation F(x(t)).

Definicije

In what follows, let y be a dependent variable and x an independent variable, and y = f(x) is an unknown function of x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the Leibniz's notation (dy/dx,d²y/dx²,...,dⁿy/dxⁿ) is more useful for differentiation and integration, whereas Lagrange's notation (y′,y′′, ..., y⁽ⁿ⁾) is more useful for representing derivatives of any order compactly, and Newton's notation $({\dot {y}},{\ddot {y}},{\overset {...}{y}})$ is often used in physics for representing derivatives of low order with respect to time.

Opšta definicija

Given F, a function of x, y, and derivatives of y. Then an equation of the form

$F\left(x,y,y',\ldots ,y^{(n-1)}\right)=y^{(n)}$

is called an explicit ordinary differential equation of order n.

More generally, an implicit ordinary differential equation of order n takes the form:

$F\left(x,y,y',y'',\ \ldots ,\ y^{(n)}\right)=0$

There are further classifications:

Autonomous

A differential equation not depending on x is called autonomous.

Linear

A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:

$y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)$

where a_i(x) and r(x) are continuous functions in x. The function r(x) is called the source term, leading to two further important classifications:

Homogeneous

If r(x) = 0, and consequently one "automatic" solution is the trivial solution, y = 0. The solution of a linear homogeneous equation is a complementary function, denoted here by y_c.

Nonhomogeneous (or inhomogeneous)

If r(x) ≠ 0. The additional solution to the complementary function is the particular integral, denoted here by y_p.

The general solution to a linear equation can be written as y = y_c + y_p.

Non-linear

A differential equation that cannot be written in the form of a linear combination.

Sistem običnih diferencijalnih jednačina

A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y₁(x), y₂(x),..., y_m(x)], and F is a vector-valued function of y and its derivatives, then

$\mathbf {y} ^{(n)}=\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)$

is an explicit system of ordinary differential equations of order n and dimension m. In column vector form:

${\begin{pmatrix}y_{1}^{(n)}\\y_{2}^{(n)}\\\vdots \\y_{m}^{(n)}\end{pmatrix}}={\begin{pmatrix}f_{1}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\f_{2}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\\vdots \\f_{m}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\end{pmatrix}}$

These are not necessarily linear. The implicit analogue is:

$\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)}\right)={\boldsymbol {0}}$

where 0 = (0, 0, ..., 0) is the zero vector. In matrix form

${\begin{pmatrix}f_{1}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\f_{2}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\\vdots \\f_{m}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}$

For a system of the form $\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}$, some sources also require that the Jacobian matrix ${\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}$ be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be [and usually is] rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.

The behavior of a system of ODEs can be visualized through the use of a phase portrait.

Rešenja

Given a differential equation

$F\left(x,y,y',\ldots ,y^{(n)}\right)=0$

a function u: I ⊂ R → R is called a solution or integral curve for F, if u is n-times differentiable on I, and

$F(x,u,u',\ \ldots ,\ u^{(n)})=0\quad x\in I.$

Given two solutions u: J ⊂ R → R and v: I ⊂ R → R, u is called an extension of v if I ⊂ J and

$u(x)=v(x)\quad x\in I.\,$

A solution that has no extension is called a maximal solution. A solution defined on all of R is called a global solution.

A general solution of an nth-order equation is a solution containing n arbitrary independent constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.

Literatura

• Halliday, David; Resnick, Robert (1977), Physics (3rd изд.), New York: Wiley, ISBN 0-471-71716-9 
• Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9 
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• Polyanin, A. D. and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
• Simmons, George F. (1972), Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, LCCN 75173716 
• Tipler, Paul A. (1991), Physics for Scientists and Engineers: Extended version (3rd изд.), New York: Worth Publishers, ISBN 0-87901-432-6 
• Boscain, Ugo; Chitour, Yacine (2011), Introduction à l'automatique (PDF) (на језику: french) CS1 одржавање: Непрепознат језик (веза)
• Dresner, Lawrence (1999), Applications of Lie's Theory of Ordinary and Partial Differential Equations, Bristol and Philadelphia: Institute of Physics Publishing, ISBN 978-0750305303 
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• Hartman, Philip (2002) [1964], Ordinary differential equations, Classics in Applied Mathematics, 38, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-510-1, MR 1929104, doi:10.1137/1.9780898719222 
• W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
• Ince, Edward L. (1944) [1926], Ordinary Differential Equations, Dover Publications, New York, ISBN 978-0-486-60349-0, MR 0010757 
• Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0-486-49510-8
• Ibragimov, Nail H (1993). CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3. Providence: CRC-Press. ISBN 0-8493-4488-3 .
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. 
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.

Spoljašnje veze

• Hazewinkel Michiel, ур. (2001). „Differential equation, ordinary”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Differential Equations на сајту Curlie (includes a list of software for solving differential equations).
• EqWorld: The World of Mathematical Equations, containing a list of ordinary differential equations with their solutions.
• Online Notes / Differential Equations by Paul Dawkins, Lamar University.
• Differential Equations, S.O.S. Mathematics.
• A primer on analytical solution of differential equations from the Holistic Numerical Methods Institute, University of South Florida.
• Ordinary Differential Equations and Dynamical Systems lecture notes by Gerald Teschl.
• Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC.
• Modeling with ODEs using Scilab A tutorial on how to model a physical system described by ODE using Scilab standard programming language by Openeering team.
• Alpha