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General solution

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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

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

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 \ref{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 \ref{2} and the $n$ relations obtained by differentiating \ref{2} with respect to $t$.

For an ordinary differential equation of order $n$,

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

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

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

from which, by an appropriate choice of the parameters, any solution of \ref{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 \ref{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 \ref{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 \ref{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://www.encyclopediaofmath.org/index.php?title=General_solution&oldid=32652
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article