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''of a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437201.png" /> ordinary differential equations
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{{TEX|done}}
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''of a system of $n$ ordinary differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437202.png" /></td> </tr></table>
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$$x'=f(t,x),\quad x=(x_1,\ldots,x_n)\in\mathbf R^n,$$
  
in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437203.png" />''
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in a domain $G$''
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437204.png" />-parameter family of vector functions
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An $n$-parameter family of vector functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$x=\phi(t,C_1,\ldots,C_n),\quad (C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{2}\tag{2}$$
  
smooth with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437206.png" />, and continuous in the parameters, from which any solution of the system can be obtained by an appropriate choice of the values of the parameters, and the graph of which is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437207.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437208.png" /> is a domain in which the conditions for the existence and uniqueness theorem for
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smooth with respect to $t$, and continuous in the parameters, from which any solution of the system can be obtained by an appropriate choice of the values of the parameters, and the graph of which is in $G\subset D$. Here $D\subset\mathbf R^{n+1}$ is a domain in which the conditions for the existence and uniqueness theorem for
  
are satisfied. (Sometimes it is agreed that the parameters may also take the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g0437209.png" />.) Geometrically the general solution of
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are satisfied. (Sometimes it is agreed that the parameters may also take the values $\pm\infty$.) Geometrically the general solution of
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372010.png" /> represents a family of non-intersecting integral curves of the system completely covering the whole domain.
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in $G$ represents a family of non-intersecting integral curves of the system completely covering the whole domain.
  
 
The general solution of
 
The general solution of
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372011.png" /> enables one to solve the [[Cauchy problem|Cauchy problem]] for the system with initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372013.png" />: The values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372014.png" /> parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372015.png" /> can be determined from the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372016.png" /> equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372017.png" />, and substituted in (2). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372018.png" /> is the solution of
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in $G$ enables one to solve the [[Cauchy problem|Cauchy problem]] for the system with initial conditions $x(t_0)=x^0$, $(t_0,x^0)\in G$: The values of the $n$ parameters $C_1,\ldots,C_n$ can be determined from the system of $n$ equations $x^0=\phi(t_0,C_1,\ldots,C_n)$, and substituted in \eqref{2}. If $x=\psi(t,t_0,x^0)$ is the solution of
  
satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372020.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372021.png" />-parameter family
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satisfying the condition $x(t_0)=x^0$, $(t_0,x^0)\in D$, then the $n$-parameter family
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372022.png" /></td> </tr></table>
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$$x=\psi(t,t_0,x_1^0,\ldots,x_n^0),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372023.png" /> is a fixed number, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372024.png" /> are regarded as parameters, is the general solution of
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where $t_0$ is a fixed number, and $x_1^0,\ldots,x_n^0$ are regarded as parameters, is the general solution of
  
in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372025.png" />, and is called the Cauchy form of the general solution. Knowing the general solution enables one to reconstruct the system of differential equations uniquely: This can be done by eliminating the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372026.png" /> parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372027.png" /> from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372028.png" /> relations (2) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372029.png" /> relations obtained by differentiating (2) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372030.png" />.
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in a domain $G\subset D$, and is called the Cauchy form of the general solution. Knowing the general solution enables one to reconstruct the system of differential equations uniquely: This can be done by eliminating the $n$ parameters $C_1,\ldots,C_n$ from the $n$ relations \eqref{2} and the $n$ relations obtained by differentiating \eqref{2} with respect to $t$.
  
For an ordinary differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372031.png" />,
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For an ordinary differential equation of order $n$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$y^{(n)}=f(x,y,y',\ldots,y^{(n-1)}),\label{3}\tag{3}$$
  
the general solution in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372033.png" /> has the form of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372034.png" />-parameter family of functions
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the general solution in a domain $G$ has the form of an $n$-parameter family of functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
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$$y=\phi(x,C_1,\ldots,C_n),\quad(C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{4}\tag{4}$$
  
from which, by an appropriate choice of the parameters, any solution of (3) can be obtained for arbitrary initial conditions
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from which, by an appropriate choice of the parameters, any solution of \eqref{3} can be obtained for arbitrary initial conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372036.png" /></td> </tr></table>
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$$y(x_0)=y_0,y'(x_0)=y_0',\ldots,y^{(n-1)}(x_0)=y_0^{(n-1)},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372037.png" /></td> </tr></table>
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$$(x_0,y_0,y_0',\ldots,y_0^{(n-1)})\in G\subset D.$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372038.png" /> is a domain in which the conditions of the existence and uniqueness theorem for (3) are satisfied.
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Here $D\subset\mathbf R^{n+1}$ is a domain in which the conditions of the existence and uniqueness theorem for \eqref{3} are satisfied.
  
 
A function obtained from the general solution for specific values of the parameters is called a particular solution. The family of functions containing all the solutions of the given system (equation) in some domain cannot always be expressed as an explicit function of the independent variable. This family may turn out to be described by an implicit function, which is called the [[General integral|general integral]], or to be described in parametric form.
 
A function obtained from the general solution for specific values of the parameters is called a particular solution. The family of functions containing all the solutions of the given system (equation) in some domain cannot always be expressed as an explicit function of the independent variable. This family may turn out to be described by an implicit function, which is called the [[General integral|general integral]], or to be described in parametric form.
  
If a specific ordinary differential equation (3) can be integrated in closed form (see [[Integration of differential equations in closed form|Integration of differential equations in closed form]]), then it is often possible to obtain relations of the type (4), where the parameters arise as integration constants and are arbitrary. (It is therefore often said that the general solution of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372039.png" />-th order equation contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043720/g04372040.png" /> arbitrary constants.) However, such a relation is far from always being the general solution in the whole domain of existence and uniqueness of the solution of the Cauchy problem for the original equation.
+
If a specific ordinary differential equation \eqref{3} can be integrated in closed form (see [[Integration of differential equations in closed form|Integration of differential equations in closed form]]), then it is often possible to obtain relations of the type \eqref{4}, where the parameters arise as integration constants and are arbitrary. (It is therefore often said that the general solution of an $n$-th order equation contains $n$ arbitrary constants.) However, such a relation is far from always being the general solution in the whole domain of existence and uniqueness of the solution of the Cauchy problem for the original equation.
  
 
====References====
 
====References====

Latest revision as of 15:28, 14 February 2020

of a system of $n$ ordinary differential equations

$$x'=f(t,x),\quad x=(x_1,\ldots,x_n)\in\mathbf R^n,$$

in a domain $G$

An $n$-parameter family of vector functions

$$x=\phi(t,C_1,\ldots,C_n),\quad (C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{2}\tag{2}$$

smooth with respect to $t$, and continuous in the parameters, from which any solution of the system can be obtained by an appropriate choice of the values of the parameters, and the graph of which is in $G\subset D$. Here $D\subset\mathbf R^{n+1}$ is a domain in which the conditions for the existence and uniqueness theorem for

are satisfied. (Sometimes it is agreed that the parameters may also take the values $\pm\infty$.) Geometrically the general solution of

in $G$ represents a family of non-intersecting integral curves of the system completely covering the whole domain.

The general solution of

in $G$ enables one to solve the Cauchy problem for the system with initial conditions $x(t_0)=x^0$, $(t_0,x^0)\in G$: The values of the $n$ parameters $C_1,\ldots,C_n$ can be determined from the system of $n$ equations $x^0=\phi(t_0,C_1,\ldots,C_n)$, and substituted in \eqref{2}. If $x=\psi(t,t_0,x^0)$ is the solution of

satisfying the condition $x(t_0)=x^0$, $(t_0,x^0)\in D$, then the $n$-parameter family

$$x=\psi(t,t_0,x_1^0,\ldots,x_n^0),$$

where $t_0$ is a fixed number, and $x_1^0,\ldots,x_n^0$ are regarded as parameters, is the general solution of

in a domain $G\subset D$, and is called the Cauchy form of the general solution. Knowing the general solution enables one to reconstruct the system of differential equations uniquely: This can be done by eliminating the $n$ parameters $C_1,\ldots,C_n$ from the $n$ relations \eqref{2} and the $n$ relations obtained by differentiating \eqref{2} with respect to $t$.

For an ordinary differential equation of order $n$,

$$y^{(n)}=f(x,y,y',\ldots,y^{(n-1)}),\label{3}\tag{3}$$

the general solution in a domain $G$ has the form of an $n$-parameter family of functions

$$y=\phi(x,C_1,\ldots,C_n),\quad(C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{4}\tag{4}$$

from which, by an appropriate choice of the parameters, any solution of \eqref{3} can be obtained for arbitrary initial conditions

$$y(x_0)=y_0,y'(x_0)=y_0',\ldots,y^{(n-1)}(x_0)=y_0^{(n-1)},$$

$$(x_0,y_0,y_0',\ldots,y_0^{(n-1)})\in G\subset D.$$

Here $D\subset\mathbf R^{n+1}$ is a domain in which the conditions of the existence and uniqueness theorem for \eqref{3} are satisfied.

A function obtained from the general solution for specific values of the parameters is called a particular solution. The family of functions containing all the solutions of the given system (equation) in some domain cannot always be expressed as an explicit function of the independent variable. This family may turn out to be described by an implicit function, which is called the general integral, or to be described in parametric form.

If a specific ordinary differential equation \eqref{3} can be integrated in closed form (see Integration of differential equations in closed form), then it is often possible to obtain relations of the type \eqref{4}, where the parameters arise as integration constants and are arbitrary. (It is therefore often said that the general solution of an $n$-th order equation contains $n$ arbitrary constants.) However, such a relation is far from always being the general solution in the whole domain of existence and uniqueness of the solution of the Cauchy problem for the original equation.

References

[1] V.V. Stepanov, "A course of differential equations" , Moscow (1959) (In Russian)
[2] N.P. Erugin, "A reader for a general course in differential equations" , Minsk (1979) (In Russian)


Comments

References

[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1980)
[a2] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5
How to Cite This Entry:
General solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General_solution&oldid=13722
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article