Diferència entre revisions de la pàgina «Equació diferencial lineal»

m
traducció automàtica feta a petició de Usuari Discussió:Gomà pendent de revisió per l'usuari
m (bot preprocessant article previ a la traduccioo automatica)
m (traducció automàtica feta a petició de Usuari Discussió:Gomà pendent de revisió per l'usuari)
{{Traducció|en|Linear differential equation}}
In MATHEMATICS , a ((((( linear DIFFERENTIAL EQUATION ))))) is of the form ..
In [[mathematics]], a '''linear [[differential equation]]''' is of the form
..
 
: <math> Ly = f \ ,</math> ..
En [[matemàtiques]], un '''[[equació diferencial]] lineal''' és de la forma
..
 
where the DIFFERENTIAL OPERATOR (( L )) is a LINEAR OPERATOR , (( y )) is the unknown function (such as a function of time y(t)) , and the RIGHT HAND SIDE &fnof ; is a given function of the same nature as (( y )) (called the ((((( source term )))))). For a function dependent on time we may write the equation more expressively as ..
 
..
 
: <math> L y(t) = f(t) \ ,</math> ..
: <math> Ly = f \,</math>
and , even more precisely by bracketing ..
 
: <math> L [y(t)] = f(t) \ ,</math> ..
 
..
 
The linear operator (( L )) may be considered to be of the form <ref>. Gershenfeld 1999 , p.9 </ref>. ..
where the [[differential operator]] ''L'' is a [[linear operator]], ''y'' is the unknown function (such as a function of time y(t)), and the [[right hand side]] &fnof; is a given function of the same nature as ''y'' (called the '''source term'''). For a function dependent on time we may write the equation more expressively as
..
 
: <math>L_n(y) \equiv \frac{d^n y}{dt^n} + A_1(t)\frac{d^{n-1}y}{dt^{n-1}} + \cdots + ..
on l'[[operador diferencial]] ''L'' és un [[operador lineal]], ''y'' és la funció desconeguda (com una funció del temps y(t)), i el [[costat de mà]] CORRECTE ƒ és una funció donada de la mateixa natura com ''y'' (anomenat el '''terme de font'''). Per a una funció dependent puntualment podem escriure l'equació més expressivament com
A_{n-1}(t)\frac{dy}{dt} + A_n(t)y \ ,</math> ..
 
..
 
The linearity condition on (( L )) rules out operations such as taking the square of the DERIVATIVE of (( y )) ; but permits , for example , taking the second derivative of (( y )) . ..
 
It is convenient to rewrite this equation in an operator form ..
: <math> L y(t) = f(t) \,</math>
..
and, even more precisely by bracketing
: <math> L_n(y) \equiv \left[\ ,D^n + A_{1}(t)D^{n-1} + \cdots + A_{n-1}(t) D + A_n(t)\right] y</math> ..
 
..
i, fins i tot més precisament reforçant
where (( D )) is the differential operator (( d/dt )) (i.e. (( Dy = y' )) , (( D )) <sup>2</sup>(( y = y" ,... )) ) , and the (( A<sub>n</sub> )) are given functions. ..
: <math> L [y(t)] = f(t) \,</math>
<!-- and the source term is considered to be a function of time &fnof ;(((( t )))) .--> ..
 
..
 
Such an equation is said to have ((((( order ))))) (( n )) , the index of the highest derivative of (( y )) that is involved. <!-- (Assuming a possibly existing coefficient (( a<sub>n</sub> )) of this derivative to be non zero , it is eliminated by dividing through it. In case it can become zero , different cases must be considered separately for the analysis of the equation.)--> ..
 
..
The linear operator ''L'' may be considered to be of the form<ref>Gershenfeld 1999, p.9</ref>
A typical simple example is the linear differential equation used to model radioactive decay <ref>. Robinson 2004 , p.5 </ref>. . Let N(t) denote the number of radioactive atoms in some sample of material (such as a portion of the cloth of the SHROUD OF TURIN <ref>. Robinson 2004 , p.7 </ref>. ) at time t. Then for some constant k > 0 , ..
 
the number of radioactive atoms which decay can be modelled by ..
L'operador lineal ''L'' pot ser considerat ser de la forma<ref>Gershenfeld 1999, p.9 </ref>.
: <math> \frac{dN}{dt} = -k N\ ,</math> ..
 
..
 
..
 
If (( y )) is assumed to be a function of only one variable , one speaks about an ORDINARY DIFFERENTIAL EQUATION , else the derivatives and their coefficients must be understood as (CONTRACTED) vectors , matrices or TENSORS of higher rank , and we have a (linear) PARTIAL DIFFERENTIAL EQUATION. ..
: <math>L_n(y) \equiv \frac{d^n y}{dt^n} + A_1(t)\frac{d^{n-1}y}{dt^{n-1}} + \cdots +
..
A_{n-1}(t)\frac{dy}{dt} + A_n(t)y \,</math>
The case where &fnof ; = 0 is called a ((((( homogeneous equation ))))) and its solutions are called ((((( complementary functions ))))). It is particularly important to the solution of the general case , since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called (( particular integral and complementary function )) ). When the (( A<sub>i</sub> )) are numbers , the equation is said to have (( CONSTANT COEFFICIENTS )) . ..
 
..
 
(==) Homogeneous equations with constant coefficients (==) ..
 
..
The linearity condition on ''L'' rules out operations such as taking the square of the [[derivative]] of ''y''; but permits, for example, taking the second derivative of ''y''.
The first method of solving linear ordinary differential equations with constant coefficients is due to EULER , who realized that solutions have the form <math>e^{z x}</math> , for possibly-complex values of <math>z</math>. The exponential function is one of the few functions that keep its shape even after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero , the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus , to solve ..
 
..
La condició de linearity en ''L'' regles fora operacions com prendre el quadrat del [[derivada|derivat]] de ''y''; però prenent els permisos, per exemple, el segon derivat de ''y'' .
: <math>y^{(n)} + A_{1}y^{(n-1)} + \cdots + A_{n}y = 0</math> ..
It is convenient to rewrite this equation in an operator form
..
 
we set <math>y=e^{z x}</math> , leading to ..
És convenient reescriure aquesta equació en una forma d'operador
..
 
: <math>z^n e^{zx} + A_1 z^{n-1} e^{zx} + \cdots + A_n e^{zx} = 0.</math> ..
 
..
 
Division by (( e )) <sup>&nbsp ;(( zx )) </sup> gives the (( n )) th-order polynomial ..
: <math> L_n(y) \equiv \left[\,D^n + A_{1}(t)D^{n-1} + \cdots + A_{n-1}(t) D + A_n(t)\right] y</math>
..
 
: <math>F(z) = z^{n} + A_{1}z^{n-1} + \cdots + A_n = 0.\ ,</math> ..
 
..
 
This algebraic equation (( F )) (((( z )))) = 0 , is the ((((( characteristic equation ))))) considered later by MONGE and CAUCHY. ..
where ''D'' is the differential operator ''d/dt'' (i.e. ''Dy = y' '', ''D''<sup>2</sup>''y = y",... ''), and the ''A<sub>n</sub>'' are given functions.
..
 
Formally , the terms ..
on ''D'' és l'operador diferencial ''d/dt'' (i.e. ''y de = Dy'', ''D'' <sup>2</sup>''y = y"... '' ), i el ''A<sub>n</sub>'' són donats funcions.
..
<!-- and the source term is considered to be a function of time &fnof;(''t'').-->
: <math>y^{(k)}\quad\quad(k = 1 , 2 , \dots , n).</math> ..
 
..
<!-- i el terme de font es considera que és una funció del temps ƒ(''t'') .-->..
of the original differential equation are replaced by (( z )) <sup>(( k )) </sup>. ROOT-FINDING ALGORITHM#FINDING ROOTS OF POLYNOMIALS .3. SOLVING .3. Solving]] the polynomial gives (( n )) values of (( z )) , (( z )) <sub>1</sub> ,&nbsp ;... ,&nbsp ;(( z )) <sub>(( n )) </sub>. Substitution of any of those values for (( z )) into (( e )) <sup>&nbsp ;(( zx )) </sup> gives a solution (( e )) <sup>&nbsp ;(( z )) <sub>(( i )) </sub>(( x )) </sup>. Since homogeneous linear differential equations obey the SUPERPOSITION PRINCIPLE , any LINEAR COMBINATION of these functions also satisfies the differential equation. ..
 
..
 
When these roots are all DISTINCT , we have (( n )) distinct solutions to the differential equation. It can be shown that these are LINEARLY INDEPENDENT , by applying the VANDERMONDE DETERMINANT , and together they form a BASIS of the space of all solutions of the differential equation. ..
 
..
Such an equation is said to have '''order''' ''n'', the index of the highest derivative of ''y'' that is involved. <!-- (Assuming a possibly existing coefficient ''a<sub>n</sub>'' of this derivative to be non zero, it is eliminated by dividing through it. In case it can become zero, different cases must be considered separately for the analysis of the equation.)-->
{{ ExampleSidebar .3. 35 % .3. ..
 
: <math>y((((( '-2y )))))+2y''-2y'+y=0 \ ,</math> ..
Tal equació es diu que té '''ordre''' ''n'', l'índex del derivat més alt de ''y'' que està implicat. <!-- (Assumint un possiblement coeficient existent ''a<sub>n</sub>'' d'aquest derivat per no ser no zero, s'elimina dividint-se a través d'això. En cas que pugui convenir a zero, els casos diferents s'han de considerar separadament per a l'anàlisi de l'equation.)-->..
has the characteristic equation ..
 
..
 
: <math>z^4-2z^3+2z^2-2z+1=0. \ ,</math> ..
 
..
A typical simple example is the linear differential equation used to model radioactive decay<ref>Robinson 2004, p.5</ref>. Let N(t) denote the number of radioactive atoms in some sample of material (such as a portion of the cloth of the [[Shroud of Turin]]<ref>Robinson 2004, p.7</ref>) at time t. Then for some constant k > 0,
This has zeroes , (( i )) , &minus ;(( i )) , and 1 (multiplicity 2). The solution basis is then ..
 
..
Un exemple simple típic és l'equació diferencial lineal fa servirda per imitar la decadència radioactiva<ref>Robinson 2004, p.5</ref>. Deixi N(t) denotar el nombre d'àtoms radioactius en alguna mostra de material (com una porció del drap de l'[[Sant sudari|Amortallar De Torí]]<ref>Robinson 2004, p.7</ref>) a t d'hora. Llavors per a alguna constant k > 0
: <math>e^{ix} ,\ , e^{-ix} ,\ , e^x ,\ , xe^x \ ,.</math> ..
the number of radioactive atoms which decay can be modelled by
..
 
This corresponds to the real-valued solution basis ..
el nombre d'àtoms radioactius que es podreixen es pot imitar a prop
..
: <math>\cos x ,\frac{dN}{dt} ,= \sin-k x ,N\ , e^x ,\ , xe^x \ ,.</math>}} ..
 
..
 
The preceding gave a solution for the case when all zeros are distinct , that is , each has MULTIPLICITY#MULTIPLICITY OF A ROOT OF A POLYNOMIAL .3. MULTIPLICITY .3. multiplicity]] 1. For the general case , if (( z )) is a (possibly complex) ZERO (or root) of (( F )) (((( z )))) having multiplicity (( m )) , then , for <math>k\in\{0 ,1 ,\dots ,m-1\} \ ,</math> , <math>y=x^ke^{zx} \ ,</math> is a solution of the Ode. Applying this to all roots gives a collection of (( n )) distinct and linearly independent functions , where (( n )) is the degree of (( F )) (((( z )))) . As before , these functions make up a basis of the solution space. ..
 
..
 
If the coefficients (( A<sub>i</sub> )) of the differential equation are real , then real-valued solutions are generally preferable. Since non-real roots (( z )) then come in CONJUGATE pairs , so do their corresponding basis functions {{ nowrap .3. (( x )) <sup>(( k )) </sup>e<sup>(( zx )) </sup>}} , and the desired result is obtained by replacing each pair with their real-valued LINEAR COMBINATIONS RE(((( Y )))) and IM(((( Y )))) , where (( y )) is one of the pair. ..
 
..
 
A case that involves complex roots can be solved with the aid of EULER'S FORMULA. ..
If ''y'' is assumed to be a function of only one variable, one speaks about an [[ordinary differential equation]], else the derivatives and their coefficients must be understood as ([[tensor contraction|contracted]]) vectors, matrices or [[tensor]]s of higher rank, and we have a (linear) [[partial differential equation]].
..
 
(===) Examples (===) ..
Si ''y'' és suposat ser una funció de només una variable, un parla sobre una [[equació diferencial ordinària|equació diferencial corrent]], més els derivats i els seus coeficients s'han d'entendre com [[(contret)]] vectors, matrius o [[tensor|tensors]] de fila més alta, i tenim un (lineal) [[equació diferencial en derivades parcials|equació diferencial parcial]].
..
 
Given <math>y(( -4y'+5y=0 \ ,</math>. The characteristic equation is <math>z^2-4z+5=0 \ ,</math> which has zeroes 2+ )) i(( and 2&#8722; )) i(( . Thus the solution basis <math>\{y_1 ,y_2\}</math> is <math>\{e^{(2+i)x} ,e^{(2-i)x}\} \ ,</math>. Now )) y'' is a solution IF AND ONLY IF <math>y=c_1y_1+c_2y_2 \ ,</math> for <math>c_1 ,c_2\in\mathbb C</math>. ..
 
..
 
Because the coefficients are real , ..
The case where &fnof; = 0 is called a '''homogeneous equation''' and its solutions are called '''complementary functions'''. It is particularly important to the solution of the general case, since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called ''particular integral and complementary function''). When the ''A<sub>i</sub>'' are numbers, the equation is said to have ''[[constant coefficients]]''.
* we are likely not interested in the complex solutions ..
 
* our basis elements are mutual conjugates ..
El cas on ƒ = 0 s'anomena un '''equació homogènia''' i les seves solucions es criden '''funcions complementàries'''. És especialment important per la solució del cas general, ja que qualsevol funció complementària es pot afegir a una solució de l'equació inhomogeneous per donar una altra solució (per un mètode tradicionalment anomenat ''funció integral i complementària particular'' ). Quan el ''A<sub>i</sub>'' són nombres, l'equació es diu que té ''[[coeficients constants]]'' .
The linear combinations ..
 
..
 
: <math>u_1=\mbox{Re}(y_1)=\frac{y_1+y_2}{2}=e^{2x}\cos(x) \ ,</math> and ..
 
..
==Homogeneous equations with constant coefficients==
: <math>u_2=\mbox{Im}(y_1)=\frac{y_1-y_2}{2i}=e^{2x}\sin(x) \ ,</math> ..
 
..
== equacions Homogènies amb coeficients constants ==
will give us a real basis in <math>\{u_1 ,u_2\}</math>. ..
 
..
 
(====) Simple harmonic oscillator (====) ..
 
..
The first method of solving linear ordinary differential equations with constant coefficients is due to [[Euler]], who realized that solutions have the form <math>e^{z x}</math>, for possibly-complex values of <math>z</math>. The exponential function is one of the few functions that keep its shape even after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus, to solve
The second order differential equation ..
 
..
El primer mètode de resoldre equacions diferencials corrents lineals amb coeficients constants és a causa d'[[Leonhard Euler|Euler]], que s'adonava que les solucions tenen la forma <math>e^{z x}</math>, per a valors possiblement complexos de <math>z</math>. La funció exponencial és una de les poques funcions que es queden la seva forma fins i tot després de diferenciació. En ordre per a la suma de derivats múltiples d'una funció per resumir a zero, els derivats s'han de neutralitzar l'un a l'altre i l'única manera que facin això és que els derivats tenen la mateixa forma com la funció inicial. Així, resoldre
: <math> D^2 y = -k^2 y , </math> ..
 
..
 
which represents a simple HARMONIC OSCILLATOR , can be restated as ..
 
..
: <math> (Dy^2{(n)} + kA_{1}y^2{(n-1)} + \cdots + A_{n}y = 0. </math> ..
 
..
 
The expression in parenthesis can be factored out , yielding ..
 
..
:we set <math> (D + i k) (D - i k) y =e^{z 0 ,x}</math>, leading to ..
 
..
posem <math>y=e^{z x}</math>, anant al davant a
which has a pair of linearly independent solutions , one for ..
 
..
 
: <math> (D - i k) y = 0 </math> ..
 
..
:<math>z^n e^{zx} + A_1 z^{n-1} e^{zx} + \cdots + A_n e^{zx} = 0.</math>
and another for ..
 
..
 
: <math> (D + i k) y = 0. </math> ..
 
..
Division by ''e''<sup>&nbsp;''zx''</sup> gives the ''n''th-order polynomial
The solutions are , respectively , ..
 
..
Divisió per ''e'' <sup> ''zx'' </sup> dóna el ''n'' polinomi de th-order
: <math> y_0 = A_0 e^{i k x} </math> ..
 
..
 
and ..
 
..
: <math> y_1F(z) = A_1z^{n} e+ A_{1}z^{n-i1} k+ x}.\cdots + A_n = 0.\,</math> ..
 
..
 
These solutions provide a basis for the two-dimensional "SOLUTION SPACE" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular , the following solutions can be constructed ..
 
..
This algebraic equation ''F''(''z'') = 0, is the '''characteristic equation''' considered later by [[Gaspard_Monge | Monge]] and [[Cauchy]].
: <math> y_{0'} = {A_0 e^{i k x} + A_1 e^{-i k x} \over 2} = C_0 \cos (k x) </math> ..
 
..
Aquesta equació algebraica ''F'' (''z'') = 0, és el '''equació característica''' considerat posterior per[[ monge]] i [[Augustin Louis Cauchy|Cauchy]].
and ..
 
..
 
: <math> y_{1'} = {A_0 e^{i k x} - A_1 e^{-i k x} \over 2 i} = C_1 \sin (k x). </math> ..
 
..
Formally, the terms
These last two trigonometric solutions are linearly independent , so they can serve as another basis for the solution space , yielding the following general solution: ..
 
..
Formalment, els termes
: <math> y_H = C_0 \cos (k x) + C_1 \sin (k x). </math> ..
 
..
 
(====) Damped harmonic oscillator (====) ..
 
Given the equation for the damped HARMONIC OSCILLATOR: ..
:<math>y^{(k)}\quad\quad(k = 1, 2, \dots, n).</math>
..
 
: <math> \left(D^2 + {b \over m} D + \omega_0^2\right) y = 0 , </math> ..
 
..
 
the expression in parentheses can be factored out: first obtain the characteristic equation by replacing (( D )) with &lambda ;. This equation must be satisfied for all (( y )) , thus: ..
of the original differential equation are replaced by ''z''<sup>''k''</sup>. [[Root-finding algorithm#Finding roots of polynomials|Solving]] the polynomial gives ''n'' values of ''z'', ''z''<sub>1</sub>,&nbsp;...,&nbsp;''z''<sub>''n''</sub>. Substitution of any of those values for ''z'' into ''e''<sup>&nbsp;''zx''</sup> gives a solution ''e''<sup>&nbsp;''z''<sub>''i''</sub>''x''</sup>. Since homogeneous linear differential equations obey the [[superposition principle]], any [[linear combination]] of these functions also satisfies the differential equation.
..
 
: <math> \lambda^2 + {b \over m} \lambda + \omega_0^2 = 0. </math> ..
de l'equació diferencial original són reemquadratts per ''z'' <sup>''k'' </sup>. [[Arrels de algorithm#finding de descobriment d'arrel de polinomis|resolent]]|RESOLENT|Resolent]] el polinomi dóna ''n'' valors de ''z'', ''z'' <sub>1</sub>, ..., ''z'' <sub>''n'' </sub>. Substitució de qualsevol d'aquells valors per ''z'' a ''e'' <sup> ''zx'' </sup> dóna una solució ''e'' <sup> ''z'' <sub>''i'' </sub>''x'' </sup>. Ja que les equacions diferencials lineals homogènies obeeixen el [[principi superposició]], qualsevol [[combinació lineal]] d'aquestes funcions també satisfà l'equació diferencial.
..
 
Solve using the QUADRATIC FORMULA: ..
 
..
 
: <math> \lambda = {-b/m \pm \sqrt{b^2 / m^2 - 4 \omega_0^2} \over 2}. </math> ..
When these roots are all [[distinct roots|distinct]], we have ''n'' distinct solutions to the differential equation. It can be shown that these are [[linearly independent]], by applying the [[Vandermonde determinant]], and together they form a [[Basis (linear algebra)|basis]] of the space of all solutions of the differential equation.
..
 
Use these data to factor out the original differential equation: ..
Quan aquestes arrels són completament [[polinomi separable|clares]], tenim ''n'' solucions clares a l'equació diferencial. Es pot mostrar que aquests són [[independència lineal|linealment independents]], aplicant el [[Matriu de Vandermonde|Determinant de vandermonde]], i junts formen una [[Base (àlgebra)|base]] de l'espai de totes les solucions de l'equació diferencial.
..
 
: <math> \left(D + {b \over 2 m} - \sqrt{{ b^2 \over 4 m^2} - \omega_0^2} \right) \left(D + {b \over 2m} + \sqrt{{ b^2 \over 4 m^2} - \omega_0^2}\right) y = 0. </math> ..
 
..
 
This implies a pair of solutions , one corresponding to ..
{{ExampleSidebar|35%|
..
 
: <math> \left(D + {b \over 2 m} - \sqrt{{ b^2 \over 4 m^2} - \omega_0^2} \right) y = 0 </math> ..
{{ ExampleSidebar|35%|
..
:<math>y''''-2y'''+2y''-2y'+y=0 \,</math>
and another to ..
has the characteristic equation
..
 
: <math> \left(D + {b \over 2m} + \sqrt{{ b^2 \over 4 m^2} - \omega_0^2}\right) y = 0 </math> ..
té l'equació característica
..
 
The solutions are , respectively , ..
 
..
 
: <math> y_0 = A_0 e^{-\omega x + \sqrt{\omega^2 - \omega_0^2} x} = A_0 e^{-\omega x} e^{\sqrt{\omega^2 - \omega_0^2} x} </math> ..
: <math>z^4-2z^3+2z^2-2z+1=0. \,</math>
..
 
and ..
 
..
 
: <math> y_1 = A_1 e^{-\omega x - \sqrt{\omega^2 - \omega_0^2} x} = A_1 e^{-\omega x} e^{-\sqrt{\omega^2 - \omega_0^2} x} </math> ..
This has zeroes, ''i'', &minus;''i'', and 1 (multiplicity 2). The solution basis is then
..
 
where &omega ; = (( b )) / 2(( m )) . From this linearly independent pair of solutions can be constructed another linearly independent pair which thus serve as a basis for the two-dimensional solution space: ..
Això té zeroes, ''i'', −''i'', i 1 (multiplicitat 2). La base de solució és llavors
..
 
: <math> y_H (A_0 , A_1) (x) = \left(A_0 \sinh \sqrt{\omega^2 - \omega_0^2} x + A_1 \cosh \sqrt{\omega^2 - \omega_0^2} x\right) e^{-\omega x}. </math> ..
 
..
 
However , if .3. &omega ; .3. < .3. &omega ;<sub>0</sub> .3. then it is preferable to get rid of the consequential imaginaries , expressing the general solution as ..
: <math>e^{ix} ,\, e^{-ix} ,\, e^x ,\, xe^x \,.</math>
..
 
: <math> y_H (A_0 , A_1) (x) = \left(A_0 \sin \sqrt{\omega_0^2 - \omega^2} x + A_1 \cos \sqrt{\omega_0^2 - \omega^2} x\right) e^{-\omega x}. </math> ..
 
..
 
This latter solution corresponds to the underdamped case , whereas the former one corresponds to the overdamped case: the solutions for the underdamped case OSCILLATE whereas the solutions for the overdamped case do not. ..
This corresponds to the real-valued solution basis
..
 
(==) Nonhomogeneous equation with constant coefficients (==) ..
Això correspon a la base de solució genuïnament valorada
..
 
To obtain the solution to the ((((( non-homogeneous equation ))))) (sometimes called ((((( inhomogeneous equation )))))) , find a particular solution (( y )) <sub>(( P )) </sub>(((( x )))) by either the METHOD OF UNDETERMINED COEFFICIENTS or the METHOD OF VARIATION OF PARAMETERS ; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution. ..
 
..
 
Suppose we face ..
: <math>\cos x ,\, \sin x ,\, e^x ,\, xe^x \,.</math>}}
..
 
: <math>\frac {d^{n}y(x)} {dx^{n}} + A_{1}\frac {d^{n-1}y(x)} {dx^{n-1}} + \cdots + A_{n}y(x) = f(x).</math> ..
 
..
 
For later convenience , define the characteristic polynomial ..
The preceding gave a solution for the case when all zeros are distinct, that is, each has [[Multiplicity#Multiplicity of a root of a polynomial|multiplicity]] 1. For the general case, if ''z'' is a (possibly complex) [[root of a function|zero]] (or root) of ''F''(''z'') having multiplicity ''m'', then, for <math>k\in\{0,1,\dots,m-1\} \,</math>, <math>y=x^ke^{zx} \,</math> is a solution of the ODE. Applying this to all roots gives a collection of ''n'' distinct and linearly independent functions, where ''n'' is the degree of ''F''(''z''). As before, these functions make up a basis of the solution space.
..
 
: <math>P(v)=v^n+A_1v^{n-1}+\cdots+A_n.</math> ..
El precedir donava una solució per al cas quan tot posa a zero són clar, és a dir, cada un té [[Multiplicity#multiplicity d'una arrel d'un polinomi|multiplicitat]]|MULTIPLICITAT|multiplicitat]] 1. Per al cas general, si ''z'' és un (possiblement complex) [[arrel aritmètica|zero]] (o arrel) de ''F'' (''z'') multiplicitat que té ''m'', llavors, per <math>k\in\{0,1,\dots,m-1\} \,</math>, <math>y=x^ke^{zx} \,</math> és una solució de l'Oda. Aplicant-se això a totes les arrels dóna una recollida de ''n'' funcions clares i linealment independents, on ''n'' és el grau de ''F'' (''z'') . Tan abans, aquestes funcions constitueixen una base per l'espai de solució.
..
 
We find the solution basis <math>\{y_1(x) ,y_2(x) ,\ldots ,y_n(x)\}</math> as in the homogeneous (((( f(x)=0 )))) case. We now seek a ((((( particular solution ))))) (( y<sub>p</sub>(x) )) by the ((((( variation of parameters ))))) method. Let the coefficients of the linear combination be functions of (( x )) : ..
 
..
 
: <math>y_p(x) = u_1(x) y_1(x) + u_2(x) y_2(x) + \cdots + u_n(x) y_n(x).</math> ..
If the coefficients ''A<sub>i</sub>'' of the differential equation are real, then real-valued solutions are generally preferable. Since non-real roots ''z'' then come in [[complex conjugate|conjugate]] pairs, so do their corresponding basis functions {{nowrap|''x''<sup>''k''</sup>e<sup>''zx''</sup>}}, and the desired result is obtained by replacing each pair with their real-valued [[linear combination]]s [[real part|Re(''y'')]] and [[Imaginary part|Im(''y'')]], where ''y'' is one of the pair.
..
 
For ease of notation we will drop the dependency on (( x )) (i.e. the various ((( x ))) ). Using the "operator" notation <math>D=d/dx</math> and a broad-minded use of notation , the Ode in question is <math>P(D)y=f</math> ; so ..
Si els coeficients ''A<sub>i</sub>'' de l'equació diferencial són real, llavors les solucions genuïnament valorades són generalment preferibles. Des d'arrels no genuïnes ''z'' llavors venir en parells de [[conjugat|conjugate]], així fa les seves funcions de base corresponents {{nowrap|''x''<sup>''k''</sup>e<sup>''zx''</sup>}}, i el resultat desitjat s'obté canviant cada parell pel seu Y de [[combinació lineal|linear combinations]] RE(''genuïnament valorat'') i Y de IM(''''), on ''y'' és un del parell.
..
 
: <math>f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\cdots+P(D)(u_ny_n).</math> ..
 
..
 
With the constraints ..
A case that involves complex roots can be solved with the aid of [[Euler's formula]].
..
 
: <math>0=u'_1y_1+u'_2y_2+\cdots+u'_ny_n</math> ..
Un cas que implica arrels complexes es pot resoldre amb l'ajut de [[Fórmula d'Euler|La fórmula d'euler]].
: <math>0=u'_1y'_1+u'_2y'_2+\cdots+u'_ny'_n</math> ..
 
: <math> \cdots</math> ..
 
: <math>0=u'_1y^{(n-2)}_1+u'_2y^{(n-2)}_2+\cdots+u'_ny^{(n-2)}_n</math> ..
 
..
===Examples===
the parameters commute out , with a little "dirt": ..
 
..
=== Exemples ===
: <math>f=u_1P(D)y_1+u_2P(D)y_2+\cdots+u_nP(D)y_n+u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.</math> ..
 
..
 
But <math>P(D)y_j=0</math> , therefore ..
 
..
Given <math>y''-4y'+5y=0 \,</math>. The characteristic equation is <math>z^2-4z+5=0 \,</math> which has zeroes 2+''i'' and 2−''i''. Thus the solution basis <math>\{y_1,y_2\}</math> is <math>\{e^{(2+i)x},e^{(2-i)x}\} \,</math>. Now ''y'' is a solution [[if and only if]] <math>y=c_1y_1+c_2y_2 \,</math> for <math>c_1,c_2\in\mathbb C</math>.
: <math>f=u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.</math> ..
 
..
Donat <math>y''-4y'+5y=0 \,</math>. L'equació característica és <math>z^2-4z+5=0 \,</math> que té zeroes 2+'' i''i 2−'' i''. Així la base de solució <math>\{y_1,y_2\}</math> és <math>\{e^{(2+i)x},e^{(2-i)x}\} \,</math>. Ara'' y '' és una solució [[si i només si]] <math>y=c_1y_1+c_2y_2 \,</math> per a <math>c_1,c_2\in\mathbb C</math>.
This , with the constraints , gives a linear system in the <math>u'_j</math>. This much can always be solved ; in fact , combining CRAMER'S RULE with the WRONSKIAN , ..
 
..
 
: <math>u'_j=(-1)^{n+j}\frac{W(y_1 ,\ldots ,y_{j-1} ,y_{j+1}\ldots ,y_n)_{0 \choose f}}{W(y_1 ,y_2 ,\ldots ,y_n)}.</math> <!-- caution: check my sign --> ..
 
..
Because the coefficients are real,
The rest is a matter of integrating <math>u'_j.</math> ..
 
..
Perquè els coeficients són genuïns
The particular solution is not unique ; <math>y_p+c_1y_1+\cdots+c_ny_n</math> also satisfies the Ode for any set of constants (( c<sub>j</sub> )) . ..
*we are likely not interested in the complex solutions
..
 
(===) Example (===) ..
* no se'ns interessa probablement en les solucions complexes
Suppose <math>y''-4y'+5y=sin(kx)</math>. We take the solution basis found above <math>\{e^{(2+i)x} ,e^{(2-i)x}\}</math>. ..
*our basis elements are mutual conjugates
: { .3. ..
 
.3. - ..
* els nostres elements de base són mutus conjuga
.3. <math>W\ ,</math> ..
The linear combinations
.3. <math>= \begin{vmatrix}e^{(2+i)x}&e^{(2-i)x} \\ (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x} \end{vmatrix}</math> ..
 
.3. - ..
Les combinacions lineals
.3. ..
 
.3. <math>=e^{4x}\begin{vmatrix}1&1\\ 2+i&2-i\end{vmatrix}</math> ..
 
.3. - ..
 
.3. ..
.3. :<math>u_1=-2ie\mbox{Re}(y_1)=\frac{y_1+y_2}{2}=e^{4x2x}\cos(x) \,</math> ..and
 
.3. } ..
 
..
 
: { .3. ..
:<math>u_2=\mbox{Im}(y_1)=\frac{y_1-y_2}{2i}=e^{2x}\sin(x) \,</math>
.3. - ..
 
.3. <math>u'_1\ ,</math> ..
 
.3. <math>=\frac{1}{W}\begin{vmatrix}0&e^{(2-i)x}\\ \sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}</math> ..
 
.3. - ..
will give us a real basis in <math>\{u_1,u_2\}</math>.
.3. ..
 
.3. <math>=-\frac{i}2\sin(kx)e^{(-2-i)x}</math> ..
donarà nosaltres una base genuïna en <math>\{u_1,u_2\}</math>.
.3. } ..
 
..
 
: { .3. ..
 
.3. - ..
==== Simple harmonic oscillator ====
.3. <math>u'_2\ ,</math> ..
 
.3. <math>=\frac{1}{W}\begin{vmatrix}e^{(2+i)x}&0\\ (2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}</math> ..
==== oscil·lador harmònic Simple ====
.3. - ..
 
.3. ..
 
.3. <math> =\frac{i}{2}\sin(kx)e^{(-2+i)x}.</math> ..
 
.3. } ..
The second order differential equation
..
 
Using the LIST OF INTEGRALS OF EXPONENTIAL FUNCTIONS ..
El segon demanar equació diferencial
..
 
: { .3. ..
 
.3. - ..
 
.3. <math>u_1\ ,</math> ..
.3. :<math> D^2 y = -\frac{i}{2}\int\sin(kx)ek^{(-2-i)x}\ y,dx </math> ..
 
.3. - ..
 
.3. ..
 
.3. <math>=\frac{ie^{(-2-i)x}}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right)</math> ..
which represents a simple [[harmonic oscillator]], can be restated as
.3. } ..
 
..
quin representa un [[moviment harmònic|oscil·lador harmònic]] simple, pot ser reafirmat com
: { .3. ..
 
.3. - ..
 
.3. <math>u_2\ ,</math> ..
 
.3. <math>=\frac i2\int\sin(kx)e^{(-2+i)x}\ ,dx</math> ..
:<math> (D^2 + k^2) y = 0. </math>
.3. - ..
 
.3. ..
 
.3. <math>=\frac{ie^{(i-2)x}}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right).</math> ..
 
.3. } ..
The expression in parenthesis can be factored out, yielding
..
 
And so ..
L'expressió en el parèntesi pot ser factored out fora, cedint
: { .3. ..
 
.3. - ..
 
.3. <math>y_p\ ,</math> ..
 
.3. <math>=\frac{i}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right) ..
:<math> (D + i k) (D - i k) y = 0,</math>
+\frac{i}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right)</math> ..
 
.3. - ..
 
.3. ..
 
.3. <math>=\frac{(5-k^2)\sin(kx)+4k\cos(kx)}{(3+k^2)^2+16}.</math> ..
which has a pair of linearly independent solutions, one for
.3. } ..
 
(Notice that (( u )) <sub>1</sub> and (( u )) <sub>2</sub> had factors that canceled (( y )) <sub>1</sub> and (( y )) <sub>2</sub> ; that is typical.) ..
quin té un parell de solucions linealment independents, un per
..
 
For interest's sake , this Ode has a physical interpretation as a driven damped HARMONIC OSCILLATOR ; (( y<sub>p</sub> )) represents the steady state , and <math>c_1y_1+c_2y_2</math> is the transient. ..
 
..
 
== Equation with variable coefficients== ..
:<math> (D - i k) y = 0 </math>
..
 
A linear Ode of order (( n )) with variable coefficients has the general form ..
 
: <math>p_{n}(x)y^{(n)}(x) + p_{n-1}(x) y^{(n-1)}(x) + \cdots + p_0(x) y(x) = r(x).</math> ..
 
..
and another for
(===) Examples (===) ..
 
..
i un altre per
A simple example is the CAUCHY–EULER EQUATION often used in engineering ..
 
..
 
: <math>x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \cdots + a_0 y(x) = 0.</math> ..
 
..
:<math> (D + i k) y = 0. </math>
(==) First order equation (==) ..
 
{{ ExampleSidebar .3. 35 % .3. Solve the equation ..
 
..
 
: <math>y'(x)+3y(x)=2 \ ,</math> ..
The solutions are, respectively,
..
 
with the initial condition ..
Les solucions són, respectivament
..
 
: <math>y\left(0\right)=2. \ ,</math> ..
 
..
 
Using the general solution method: ..
:<math> y_0 = A_0 e^{i k x} </math>
..
 
: <math>y=e^{-3x}\left(\int 2 e^{3x}\ , dx + \kappa\right). \ ,</math> ..
 
..
 
The indefinite integral is solved to give: ..
and
..
 
: <math>y=e^{-3x}\left(2/3 e^{3x} + \kappa\right). \ ,</math> ..
i
..
 
Then we can reduce to: ..
 
..
 
: <math>y=2/3 + \kappa e^{-3x}. \ ,</math> ..
:<math> y_1 = A_1 e^{-i k x}. </math>
..
 
where (( &kappa ; )) is 4/3 from the initial condition.}} ..
 
A linear Ode of order 1 with variable coefficients has the general form ..
 
..
These solutions provide a basis for the two-dimensional "[[vector space|solution space]]" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular, the following solutions can be constructed
: <math>Dy(x) + f(x) y(x) = g(x).</math> ..
 
..
Aquestes solucions proporcionen una base per l'"[[espai vectorial|espai de solució]]" bidimensional del segon ordre equació diferencial: significant que les combinacions lineals d'aquestes solucions també seran solucions. En particular, les solucions següents es poden construir
Equations of this form can be solved by multiplying the INTEGRATING FACTOR ..
 
..
 
: <math>e^{\int f(x)\ ,dx}</math> ..
 
..
:<math> y_{0'} = {A_0 e^{i k x} + A_1 e^{-i k x} \over 2} = C_0 \cos (k x) </math>
throughout to obtain ..
 
..
 
: <math> Dy(x)e^{\int f(x)\ ,dx}+f(x)y(x)e^{\int f(x)\ ,dx}=g(x)e^{\int f(x) \ , dx} ,</math> ..
 
..
and
which simplifies due to the PRODUCT RULE to ..
 
..
i
: <math> D (y(x)e^{\int f(x)\ ,dx})=g(x)e^{\int f(x)\ ,dx}</math> ..
 
..
 
which , on integrating both sides , yields ..
 
..
: <math> y(x)y_{1'} = {A_0 e^{\inti f(x)\k ,dxx}=\int g(x)- A_1 e^{\int-i k f(x)} \over 2 ,dxi} = C_1 \sin ,dx+c(k ~x). ,</math> ..
 
..
 
: <math> y(x) = {\int g(x)e^{\int f(x)\ ,dx} \ ,dx+c \over e^{\int f(x)\ ,dx}} ~.</math> ..
 
..
These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:
In other words: The solution of a first-order linear Ode ..
 
..
Aquestes dues últimes solucions trigonometric són linealment independents, així poden servir d'una altra base per l'espai de solució, produint la solució general següent:
: <math>y'(x) + f(x) y(x) = g(x) ,</math> ..
 
..
 
with coefficients that may or may not vary with (( x )) , is: ..
 
..
: <math>y y_H =e^{-a(x)} C_0 \left(\intcos g(x)k e^{a(x)}\ ,+ dxC_1 +\sin \kappa\right(k x). </math> ..
 
..
 
where (( <math>\kappa</math> )) is the constant of integration , and ..
 
..
==== Damped harmonic oscillator ====
: <math>a(x)=\int{f(x)\ ,dx}.</math> ..
 
..
==== oscil·lador harmònic Humitejat ====
(===) Examples (===) ..
Given the equation for the damped [[harmonic oscillator]]:
Consider a first order differential equation with CONSTANT COEFFICIENTS: ..
 
..
Donat l'equació per a l'[[moviment harmònic|oscil·lador harmònic]] humitejat:
: <math>\frac{dy}{dx} + b y = 1.</math> ..
 
..
 
This equation is particularly relevant to first order systems such as RC CIRCUITS and MASS-DAMPER systems. ..
 
..
:<math> \left(D^2 + {b \over m} D + \omega_0^2\right) y = 0, </math>
In this case , (( p )) (((( x )))) = b , (( r )) (((( x )))) = 1. ..
 
..
 
Hence its solution is ..
 
..
the expression in parentheses can be factored out: first obtain the characteristic equation by replacing ''D'' with &lambda;. This equation must be satisfied for all ''y'', thus:
: <math>y(x) = e^{-bx} \left( e^{bx}/b+ C \right) = 1/b + C e^{-bx} .</math> ..
 
..
l'expressió en parèntesis pot ser factored out fora: primer obtenir l'equació característica reemquadratnt ''D'' amb λ;. Aquesta equació ha de ser satisfeta per a tot ''y'', així:
(==) See also (==) ..
 
* CONTINUOUS-REPAYMENT_MORTGAGE#ORDINARY_TIME_DIFFERENTIAL_EQUATION .3. CONTINUOUS-REPAYMENT MORTGAGE .3. Continuous-repayment mortgage]] ..
 
* FOURIER TRANSFORM ..
 
* LAPLACE TRANSFORM ..
:<math> \lambda^2 + {b \over m} \lambda + \omega_0^2 = 0. </math>
..
 
(==) Notes (==) ..
 
{{ reflist .3. 2}} ..
 
..
Solve using the [[quadratic formula]]:
(==) References (==) ..
 
* {{ Citation ..
Resolgui fa servirnt la [[equació de segon grau|fórmula quadràtica]]:
.3. author = Birkhoff , Garret and Rota , Gian-Carlo ..
 
.3. year = 1978 ..
 
.3. title = Ordinary Differential Equations ..
 
.3. isbn = 0-471-07411-X ..
:<math> \lambda = {-b/m \pm \sqrt{b^2 / m^2 - 4 \omega_0^2} \over 2}. </math>
.3. publisher = John Wiley and Sons , Inc. ..
 
.3. location = New York ..
 
.3. oclc = ..
 
}} ..
Use these data to factor out the original differential equation:
* {{ Citation ..
 
.3. author = Gershenfeld , Neil ..
Utilitzi aquestes dades a factor fora l'equació diferencial original:
.3. year = 1999 ..
 
.3. title =The Nature of Mathematical Modeling ..
 
.3. isbn = 978-0521-570954 ..
 
.3. publisher = Cambridge University Press ..
:<math> \left(D + {b \over 2 m} - \sqrt{{b^2 \over 4 m^2} - \omega_0^2} \right) \left(D + {b \over 2m} + \sqrt{{b^2 \over 4 m^2} - \omega_0^2}\right) y = 0. </math>
.3. location = Cambridge , Uk. ..
 
.3. oclc = ..
 
}} ..
 
* {{ Citation ..
This implies a pair of solutions, one corresponding to
.3. author = Robinson , James C. ..
 
.3. year = 2004 ..
Això implica un parell de solucions, corresponent-se un a
.3. title = An Introduction to Ordinary Differential Equations ..
 
.3. isbn = 0-521-826500 ..
 
.3. publisher = Cambridge University Press ..
 
.3. location = Cambridge , Uk. ..
:<math> \left(D + {b \over 2 m} - \sqrt{{b^2 \over 4 m^2} - \omega_0^2} \right) y = 0 </math>
.3. oclc = ..
 
}} ..
 
..
 
[[Category:Differential equations]] ..
and another to
..
 
[[ar:معادلة تفاضلية خطية]] ..
i un altre a
[[cs:Lineární diferenciální rovnice]] ..
 
[[de:Lineare gewöhnliche Differentialgleichung]] ..
 
[[es:Ecuación diferencial lineal]] ..
 
[[fr:Équation différentielle linéaire]] ..
:<math> \left(D + {b \over 2m} + \sqrt{{b^2 \over 4 m^2} - \omega_0^2}\right) y = 0 </math>
[[it:Equazione differenziale lineare]] ..
 
[[he:משוואה דיפרנציאלית לינארית]] ..
 
[[ja:線型微分方程式]] ..
 
[[pt:Equação diferencial linear]] ..
The solutions are, respectively,
[[ru:Линейное дифференциальное уравнение]] ..
 
[[sv:Linjär differentialekvation]] ..
Les solucions són, respectivament
[[zh:线性微分方程]] ..
 
paraulesenllacos ..
 
..
 
MATHEMATICS ..
:<math> y_0 = A_0 e^{-\omega x + \sqrt{\omega^2 - \omega_0^2} x} = A_0 e^{-\omega x} e^{\sqrt{\omega^2 - \omega_0^2} x} </math>
..
 
DIFFERENTIAL EQUATION ..
 
..
 
DIFFERENTIAL OPERATOR ..
and
..
 
LINEAR OPERATOR ..
i
..
 
RIGHT HAND SIDE ..
 
..
 
DERIVATIVE ..
:<math> y_1 = A_1 e^{-\omega x - \sqrt{\omega^2 - \omega_0^2} x} = A_1 e^{-\omega x} e^{-\sqrt{\omega^2 - \omega_0^2} x} </math>
..
 
SHROUD OF TURIN ..
 
..
 
ORDINARY DIFFERENTIAL EQUATION ..
where &omega; = ''b'' / 2''m''. From this linearly independent pair of solutions can be constructed another linearly independent pair which thus serve as a basis for the two-dimensional solution space:
..
 
CONTRACTED ..
on ω; = ''b'' / 2 ''m'' . Des d'aquest parell linealment independent de solucions pot ser construït un altre parell linealment independent que així serveixen com a base per l'espai de solució bidimensional:
..
 
TENSORS ..
 
..
 
PARTIAL DIFFERENTIAL EQUATION ..
:<math> y_H (A_0, A_1) (x) = \left(A_0 \sinh \sqrt{\omega^2 - \omega_0^2} x + A_1 \cosh \sqrt{\omega^2 - \omega_0^2} x\right) e^{-\omega x}. </math>
..
 
CONSTANT COEFFICIENTS ..
 
..
 
EULER ..
However, if |&omega;| < |&omega;<sub>0</sub>| then it is preferable to get rid of the consequential imaginaries, expressing the general solution as
..
 
MONGE ..
Tanmateix, si|ω;| < |ω;<sub>0</sub>| llavors és preferible aconseguir lliurar dels imaginaries consequential, expressant la solució general com
..
 
CAUCHY ..
 
..
 
ROOT-FINDING ALGORITHM#FINDING ROOTS OF POLYNOMIALS .3. SOLVING ..
:<math> y_H (A_0, A_1) (x) = \left(A_0 \sin \sqrt{\omega_0^2 - \omega^2} x + A_1 \cos \sqrt{\omega_0^2 - \omega^2} x\right) e^{-\omega x}. </math>
..
 
SUPERPOSITION PRINCIPLE ..
 
..
 
LINEAR COMBINATION ..
This latter solution corresponds to the underdamped case, whereas the former one corresponds to the overdamped case: the solutions for the underdamped case [[oscillation|oscillate]] whereas the solutions for the overdamped case do not.
..
 
DISTINCT ..
Aquesta última solució correspon al cas d'underdamped, mentre que l'anterior correspon al cas sobrehumitejat: les solucions per al cas d'underdamped [[oscil·lació|oscil·len]] mentre que les solucions per al cas sobrehumitejat fan no.
..
 
LINEARLY INDEPENDENT ..
 
..
 
VANDERMONDE DETERMINANT ..
==Nonhomogeneous equation with constant coefficients==
..
 
BASIS ..
== equació No Homogènia amb coeficients constants ==
..
 
MULTIPLICITY#MULTIPLICITY OF A ROOT OF A POLYNOMIAL .3. MULTIPLICITY ..
 
..
 
ZERO ..
To obtain the solution to the '''non-homogeneous equation''' (sometimes called '''inhomogeneous equation'''), find a particular solution ''y''<sub>''P''</sub>(''x'') by either the [[method of undetermined coefficients]] or the [[method of variation of parameters]]; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution.
..
 
CONJUGATE ..
Obtenir la solució al '''equació no homogènia''' (a vegades anomenat '''equació inhomogeneous'''), troba una solució particular ''y'' <sub>''P'' </sub>(''x'') o pel [[mètode de coeficients irresoluts]] o pel [[mètode de variació dels paràmetres|mètode de variació de parameters]]; la solució general a l'equació diferencial lineal és la suma de la solució general de l'equació homogènia relacionada i la solució particular.
..
 
LINEAR COMBINATIONS ..
 
..
 
RE(((( Y )))) ..
Suppose we face
..
 
IM(((( Y )))) ..
Suposi que mirem
..
 
EULER'S FORMULA ..
 
..
 
IF AND ONLY IF ..
:<math>\frac {d^{n}y(x)} {dx^{n}} + A_{1}\frac {d^{n-1}y(x)} {dx^{n-1}} + \cdots + A_{n}y(x) = f(x).</math>
..
 
HARMONIC OSCILLATOR ..
 
..
 
SOLUTION SPACE ..
For later convenience, define the characteristic polynomial
..
 
HARMONIC OSCILLATOR ..
Per a la conveniència posterior, defineixi el polinomi característic
..
 
QUADRATIC FORMULA ..
 
..
 
OSCILLATE ..
:<math>P(v)=v^n+A_1v^{n-1}+\cdots+A_n.</math>
..
 
METHOD OF UNDETERMINED COEFFICIENTS ..
 
..
 
METHOD OF VARIATION OF PARAMETERS ..
We find the solution basis <math>\{y_1(x),y_2(x),\ldots,y_n(x)\}</math> as in the homogeneous (''f(x)=0'') case. We now seek a '''particular solution''' ''y<sub>p</sub>(x)'' by the '''variation of parameters''' method. Let the coefficients of the linear combination be functions of ''x'':
..
 
CRAMER'S RULE ..
Trobem la base de solució <math>\{y_1(x),y_2(x),\ldots,y_n(x)\}</math> com en l'homogeni (''f(x)=0'') cas. Ara busquem un '''solució particular''' ''y<sub>p</sub>(x)'' pel '''variació de paràmetres''' mètode. Deixi els coeficients de la combinació lineal que és funcions de ''x'' :
..
 
WRONSKIAN ..
 
..
 
LIST OF INTEGRALS OF EXPONENTIAL FUNCTIONS ..
:<math>y_p(x) = u_1(x) y_1(x) + u_2(x) y_2(x) + \cdots + u_n(x) y_n(x).</math>
..
 
HARMONIC OSCILLATOR ..
 
..
 
CAUCHY–EULER EQUATION ..
For ease of notation we will drop the dependency on ''x'' (i.e. the various ''(x)''). Using the "operator" notation <math>D=d/dx</math> and a broad-minded use of notation, the ODE in question is <math>P(D)y=f</math>; so
..
 
INTEGRATING FACTOR ..
Per a la facilitat de notació deixarem la dependència damunt ''x'' (i.e. els diversos ''(x)'' ). Utilitzant la notació d'"operador" <math>D=d/dx</math> i un ús amplament importat de notació, l'Oda en qüestió és <math>P(D)y=f</math>; així
..
 
PRODUCT RULE ..
 
..
 
CONSTANT COEFFICIENTS ..
:<math>f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\cdots+P(D)(u_ny_n).</math>
..
 
RC CIRCUITS ..
 
..
 
MASS-DAMPER ..
With the constraints
..
 
CONTINUOUS-REPAYMENT_MORTGAGE#ORDINARY_TIME_DIFFERENTIAL_EQUATION .3. CONTINUOUS-REPAYMENT MORTGAGE ..
Amb les coaccions
..
 
FOURIER TRANSFORM ..
 
..
 
LAPLACE TRANSFORM ..
:<math>0=u'_1y_1+u'_2y_2+\cdots+u'_ny_n</math>
:<math>0=u'_1y'_1+u'_2y'_2+\cdots+u'_ny'_n</math>
:<math> \cdots</math>
:<math>0=u'_1y^{(n-2)}_1+u'_2y^{(n-2)}_2+\cdots+u'_ny^{(n-2)}_n</math>
 
 
 
the parameters commute out, with a little "dirt":
 
els paràmetres es desplacen fora, amb una bit de "brutícia":
 
 
 
:<math>f=u_1P(D)y_1+u_2P(D)y_2+\cdots+u_nP(D)y_n+u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.</math>
 
 
 
But <math>P(D)y_j=0</math>, therefore
 
Però <math>P(D)y_j=0</math>, per això
 
 
 
:<math>f=u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.</math>
 
 
 
This, with the constraints, gives a linear system in the <math>u'_j</math>. This much can always be solved; in fact, combining [[Cramer's rule]] with the [[Wronskian]],
 
Això, amb les coaccions, dóna un sistema lineal en el <math>u'_j</math>. Tant sempre es pot resoldre; de fet, combinant [[Regla de Cramer|La regla de cramer]] amb el [[Wronskià|Wronskian]]
 
 
 
:<math>u'_j=(-1)^{n+j}\frac{W(y_1,\ldots,y_{j-1},y_{j+1}\ldots,y_n)_{0 \choose f}}{W(y_1,y_2,\ldots,y_n)}.</math> <!-- caution: check my sign -->
 
 
 
The rest is a matter of integrating <math>u'_j.</math>
 
La resta és una qüestió d'integrar <math>u'_j.</math>..
 
 
 
The particular solution is not unique; <math>y_p+c_1y_1+\cdots+c_ny_n</math> also satisfies the ODE for any set of constants ''c<sub>j</sub>''.
 
La solució particular no és única; <Math>y_p+c_1y_1+\cdots+c_ny_n</math> també satisfà l'Oda per a qualsevol conjunt de constants ''c<sub>j</sub>'' .
 
 
 
===Example===
 
=== Exemple ===
Suppose <math>y''-4y'+5y=sin(kx)</math>. We take the solution basis found above <math>\{e^{(2+i)x},e^{(2-i)x}\}</math>.
 
Suposi <math>y''-4y'+5y=sin(kx)</math>. Prenem la base de solució trobada damunt <math>\{e^{(2+i)x},e^{(2-i)x}\}</math>.
:{|
 
: {|
|-
 
|-
|<math>W\,</math>
 
|<Math>W\,</math>
|<math>= \begin{vmatrix}e^{(2+i)x}&e^{(2-i)x} \\ (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x} \end{vmatrix}</math>
 
|I)x de <math>= \begin{vmatrix}e^{(2+i)x}&e^{(2} \\ (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x} \end{vmatrix}</math>
|-
 
|-
|
 
|
|<math>=e^{4x}\begin{vmatrix}1&1\\ 2+i&2-i\end{vmatrix}</math>
 
|<Math>=e^{4x}\begin{vmatrix}1&1\\ 2+i&2-i\end{vmatrix}</math>
|-
 
|-
|
 
|
|<math>=-2ie^{4x}\,</math>
 
|<Math>=-2ie^{4x}\,</math>
|}
 
|}
 
 
 
:{|
 
: {|
|-
 
|-
|<math>u'_1\,</math>
 
|<Math>u'_1\,</math>
|<math>=\frac{1}{W}\begin{vmatrix}0&e^{(2-i)x}\\ \sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}</math>
 
|<Math>=\frac{1}{W}\begin{vmatrix}0&e^{(2-i)x}\\ \sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}</math>
|-
 
|-
|
 
|
|<math>=-\frac{i}2\sin(kx)e^{(-2-i)x}</math>
 
|<Math>=-\frac{i}2\sin(kx)e^{(-2-i)x}</math>
|}
 
|}
 
 
 
:{|
 
: {|
|-
 
|-
|<math>u'_2\,</math>
 
|<Math>u'_2\,</math>
|<math>=\frac{1}{W}\begin{vmatrix}e^{(2+i)x}&0\\ (2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}</math>
 
|<Math>=\frac{1}{W}\begin{vmatrix}e^{(2+i)x}&0\\ (2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}</math>
|-
 
|-
|
 
|
|<math> =\frac{i}{2}\sin(kx)e^{(-2+i)x}.</math>
 
|<Math> =\frac{i}{2}\sin(kx)e^{(-2+i)x}.</math>
|}
 
|}
 
 
 
Using the [[list of integrals of exponential functions]]
 
Utilitzant la [[llista d'integrals de funcions exponencials]]
 
 
 
:{|
 
: {|
|-
 
|-
|<math>u_1\,</math>
 
|<Math>u_1\,</math>
|<math>=-\frac{i}{2}\int\sin(kx)e^{(-2-i)x}\,dx</math>
 
|<Math>=-\frac{i}{2}\int\sin(kx)e^{(-2-i)x}\,dx</math>
|-
 
|-
|
 
|
|<math>=\frac{ie^{(-2-i)x}}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right)</math>
 
|<Math>=\frac{ie^{(-2-i)x}}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right)</math>
|}
 
|}
 
 
 
:{|
 
: {|
|-
 
|-
|<math>u_2\,</math>
 
|<Math>u_2\,</math>
|<math>=\frac i2\int\sin(kx)e^{(-2+i)x}\,dx</math>
 
|<Math>=\frac i2\int\sin(kx)e^{(-2+i)x}\,dx</math>
|-
 
|-
|
 
|
|<math>=\frac{ie^{(i-2)x}}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right).</math>
 
|<Math>=\frac{ie^{(i-2)x}}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right).</math>
|}
 
|}
 
 
 
And so
 
I així
:{|
 
: {|
|-
 
|-
|<math>y_p\,</math>
 
|<Math>y_p\,</math>
|<math>=\frac{i}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right)
 
|<Math>=\frac{i}{2(3+4i+k^2)}\left((2+i)\sin(kx)+k\cos(kx)\right)
+\frac{i}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right)</math>
 
+\frac{i}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right)</math>
|-
 
|-
|
 
|
|<math>=\frac{(5-k^2)\sin(kx)+4k\cos(kx)}{(3+k^2)^2+16}.</math>
 
|<Math>=\frac{(5-k^2)\sin(kx)+4k\cos(kx)}{(3+k^2)^2+16}.</math>
|}
 
|}
(Notice that ''u''<sub>1</sub> and ''u''<sub>2</sub> had factors that canceled ''y''<sub>1</sub> and ''y''<sub>2</sub>; that is typical.)
 
(Avís que ''u'' <sub>1</sub> i ''u'' <sub>2</sub> tenia factors que anul·laven ''y'' <sub>1</sub> i ''y'' <sub>2</sub>; allò és típic.)
 
 
 
For interest's sake, this ODE has a physical interpretation as a driven damped [[harmonic oscillator]]; ''y<sub>p</sub>'' represents the steady state, and <math>c_1y_1+c_2y_2</math> is the transient.
 
Per al motiu d'interès, aquesta Oda té una interpretació física com un [[moviment harmònic|oscil·lador harmònic]] humitejat conduït; ''y<sub>p</sub>'' representa el règim permanent, i <math>c_1y_1+c_2y_2</math> és el transeünt.
 
 
 
== Equation with variable coefficients==
 
Equació de == amb coefficients== variable
 
 
 
A linear ODE of order ''n'' with variable coefficients has the general form
 
Una Oda lineal d'ordre ''n'' amb coeficients variables té la forma general
:<math>p_{n}(x)y^{(n)}(x) + p_{n-1}(x) y^{(n-1)}(x) + \cdots + p_0(x) y(x) = r(x).</math>
 
 
 
===Examples===
 
=== Exemples ===
 
 
 
A simple example is the [[Cauchy–Euler equation]] often used in engineering
 
Un exemple simple és l'[[Equació D'euler]] de CAUCHY sovint fa servirda ideant
 
 
 
:<math>x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \cdots + a_0 y(x) = 0.</math>
 
 
 
== First order equation ==
 
== Primera equació d'ordre ==
{{ExampleSidebar|35%|Solve the equation
 
{{ ExampleSidebar|35%|Resolgui l'equació
 
 
 
: <math>y'(x)+3y(x)=2 \,</math>
 
 
 
with the initial condition
 
amb la condició inicial
 
 
 
: <math>y\left(0\right)=2. \,</math>
 
 
 
Using the general solution method:
 
Utilitzant el mètode de solució general:
 
 
 
: <math>y=e^{-3x}\left(\int 2 e^{3x}\, dx + \kappa\right). \,</math>
 
 
 
The indefinite integral is solved to give:
 
La integral indefinida es resol per cedir:
 
 
 
: <math>y=e^{-3x}\left(2/3 e^{3x} + \kappa\right). \,</math>
 
 
 
Then we can reduce to:
 
Llavors ens podem reduir a:
 
 
 
: <math>y=2/3 + \kappa e^{-3x}. \,</math>
 
 
 
where ''&kappa;'' is 4/3 from the initial condition.}}
 
on ''κ;'' és 4/3 de la inicial condiciona.}}
A linear ODE of order 1 with variable coefficients has the general form
 
Una Oda lineal de l'ordre 1 amb coeficients variables té la forma general
 
 
 
:<math>Dy(x) + f(x) y(x) = g(x).</math>
 
 
 
Equations of this form can be solved by multiplying the [[integrating factor]]
 
Les equacions d'aquesta forma es poden resoldre multiplicant el [[factor d'integració|factor integrant]]
 
 
 
:<math>e^{\int f(x)\,dx}</math>
 
 
 
throughout to obtain
 
per obtenir
 
 
 
:<math> Dy(x)e^{\int f(x)\,dx}+f(x)y(x)e^{\int f(x)\,dx}=g(x)e^{\int f(x) \, dx},</math>
 
 
 
which simplifies due to the [[product rule]] to
 
que simplifica a causa de la [[regla del producte|regla de producte]] a
 
 
 
: <math> D (y(x)e^{\int f(x)\,dx})=g(x)e^{\int f(x)\,dx}</math>
 
 
 
which, on integrating both sides, yields
 
que, integrant els dos costats, cedeix
 
 
 
: <math> y(x)e^{\int f(x)\,dx}=\int g(x)e^{\int f(x)\,dx} \,dx+c ~,</math>
 
 
 
: <math> y(x) = {\int g(x)e^{\int f(x)\,dx} \,dx+c \over e^{\int f(x)\,dx}} ~.</math>
 
 
 
In other words: The solution of a first-order linear ODE
 
En altres paraules: La solució d'una Oda lineal de primer ordre
 
 
 
: <math>y'(x) + f(x) y(x) = g(x),</math>
 
 
 
with coefficients that may or may not vary with ''x'', is:
 
amb coeficients allò pot o pot no variar amb ''x'', és:
 
 
 
:<math>y=e^{-a(x)}\left(\int g(x) e^{a(x)}\, dx + \kappa\right)</math>
 
 
 
where ''<math>\kappa</math>'' is the constant of integration, and
 
on ''<math>\kappa</math>'' és la constant d'integració, i
 
 
 
: <math>a(x)=\int{f(x)\,dx}.</math>
 
 
 
=== Examples ===
 
=== Exemples ===
Consider a first order differential equation with [[constant coefficients]]:
 
Consideri un primer ordre equació diferencial amb [[coeficients constants]]:
 
 
 
:<math>\frac{dy}{dx} + b y = 1.</math>
 
 
 
This equation is particularly relevant to first order systems such as [[RC circuit]]s and [[damping|mass-damper]] systems.
 
Aquesta equació és especialment pertinent a primers sistemes d'ordre com [[Rc circuits]] i sistemes més [[massius]] HUMITS.
 
 
 
In this case, ''p''(''x'') = b, ''r''(''x'') = 1.
 
En aquest cas ''pàg.'' (''x'') = b, ''r'' (''x'') = 1.
 
 
 
Hence its solution is
 
Per això la seva solució és
 
 
 
:<math>y(x) = e^{-bx} \left( e^{bx}/b+ C \right) = 1/b + C e^{-bx} .</math>
 
 
 
==See also==
 
== Vegeu també ==
* [[Continuous-repayment_mortgage#Ordinary_time_differential_equation | Continuous-repayment mortgage]]
 
* [[Repayment_mortgage#ordinary_time_differential_equation|continu Hipoteca De Reemborsament Continu]]|CONTINU HIPOTECA DE REEMBORSAMENT CONTINU| Hipoteca de reemborsament continu]]
* [[Fourier transform]]
 
* [[Transformada de Fourier| transformació de]] FOURIER
* [[Laplace transform]]
 
* [[Transformada de Laplace| transformació de]] LAPLACE
 
 
 
== Notes ==
 
== Notes ==
{{reflist|2}}
 
{{Referències|2}}
 
 
 
== References ==
 
== Referències ==
*{{Citation
 
* {{Ref-llibre
| author = Birkhoff, Garret and Rota, Gian-Carlo
 
|author = Birkhoff, Garret and Rota, Gian-Carlo
| year = 1978
 
|any = 1978
| title = Ordinary Differential Equations
 
|títol = Ordinary Differential Equations
| isbn = 0-471-07411-X
 
|isbn = 0-471-07411-X
| publisher = John Wiley and Sons, Inc.
 
|editorial = John Wiley and Sons, Inc.
| location = New York
 
|lloc = Nova York
| oclc =
 
|oclc =
}}
 
}}
*{{Citation
 
* {{Ref-llibre
| author = Gershenfeld, Neil
 
|author = Gershenfeld, Neil
| year = 1999
 
|any = 1999
| title =The Nature of Mathematical Modeling
 
|títol = The Nature of Mathematical Modeling
| isbn = 978-0521-570954
 
|isbn = 978-0521-570954
| publisher = Cambridge University Press
 
|editorial = Cambridge University Press
| location = Cambridge, UK.
 
|lloc = Cambridge, Uk.
| oclc =
 
|oclc =
}}
 
}}
*{{Citation
 
* {{Ref-llibre
| author = Robinson, James C.
 
|author = Robinson, James C.
| year = 2004
 
|any = 2004
| title = An Introduction to Ordinary Differential Equations
 
|títol = An Introduction to Ordinary Differential Equations
| isbn = 0-521-826500
 
|isbn = 0-521-826500
| publisher = Cambridge University Press
 
|editorial = Cambridge University Press
| location = Cambridge, UK.
 
|lloc = Cambridge, Uk.
| oclc =
 
|oclc =
}}
 
}}
 
 
 
[[Category:Differential equations]]
 
[[ar:معادلة تفاضلية خطية]]
[[cs:Lineární diferenciální rovnice]]
[[de:Lineare gewöhnliche Differentialgleichung]]
[[es:Ecuación diferencial lineal]]
[[fr:Équation différentielle linéaire]]
[[it:Equazione differenziale lineare]]
[[he:משוואה דיפרנציאלית לינארית]]
[[ja:線型微分方程式]]
[[pt:Equação diferencial linear]]
[[ru:Линейное дифференциальное уравнение]]
[[sv:Linjär differentialekvation]]
[[zh:线性微分方程]]
[[en:Linear differential equation]]
170.156

modificacions