Linear ordinary differential equation

Linear ordinary differential equations or systems of linear ordinary differential equations are an important class of ordinary differential equations .

definition

Linear ordinary differential equations are differential equations of form

${\displaystyle y^{(n)}(x)=\sum _{k=0}^{n-1}a_{k}(x)y^{(k)}(x)+g(x)\,,}$

in which an unknown real, complex or vector-valued function defined on an interval is sought that fulfills the equation presented. The -th derivative denotes the function sought. If the equation is equal to the null function , one speaks of a homogeneous equation, otherwise of an inhomogeneous equation. The function is also called inhomogeneity . Like the coefficient functions, it is a continuous, completely defined function. In the vector-valued case, the matrices are square and the equation represents a linear differential equation system for the components of the solution function. In the important special case that they do not depend on, the equation is called a linear differential equation with constant coefficients . ${\displaystyle I}$ ${\displaystyle y}$ ${\displaystyle y^{(k)}}$ ${\displaystyle k}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle a_{k}}$ ${\displaystyle I}$ ${\displaystyle a_{k}}$ ${\displaystyle y=(y_{1},\ldots ,y_{m})}$ ${\displaystyle a_{k}}$ ${\displaystyle x}$

An essential property of linear equations is the principle of superposition : If the equation solves with inhomogeneity and with inhomogeneity , then the linear combination solves the equation with inhomogeneity.In particular, in the homogeneous case, sums and multiples of solutions are always solutions. The reason is that a higher dissipation in a linear manner of lower derivatives depends. ${\displaystyle y(x)}$ ${\displaystyle g(x)}$ ${\displaystyle z(x)}$ ${\displaystyle h(x)}$ ${\displaystyle \alpha y(x)+\beta z(x)}$ ${\displaystyle \alpha g(x)+\beta h(x).}$ ${\displaystyle y^{(n)}}$ ${\displaystyle y,\ldots ,y^{(n-1)}}$

Examples

• The first order linear differential equation system of equations ${\displaystyle m}$

${\displaystyle \ y'=A(x)y+g(x)\ ,}$

where and are continuous functions. The corresponding homogeneous system is ${\displaystyle A\colon I\rightarrow \mathbb {R} ^{m\times m}}$ ${\displaystyle g\colon I\rightarrow \mathbb {R} ^{m}}$

${\displaystyle \ y'=A(x)y\ .}$

• The -th order linear differential equation ${\displaystyle n}$

${\displaystyle \sum _{i=0}^{n}a_{i}(x)y^{(i)}=g(x)\ ,}$

where are continuous functions. The corresponding homogeneous equation is ${\displaystyle a_{i},g\colon I\rightarrow \mathbb {R} }$

${\displaystyle \sum _{i=0}^{n}a_{i}(x)y^{(i)}=0\ .}$

The latter group also includes the following differential equations:

• Airy's differential equation

${\displaystyle \ y''-\lambda xy=0}$

• Bessel differential equation

${\displaystyle \ x^{2}y''+xy'+(x^{2}-n^{2})y=0,\ n\in \mathbb {R} }$.

• Euler's differential equation

${\displaystyle \sum _{i=0}^{n}b_{i}(cx+d)^{i}y^{(i)}(x)=0}$

• Hermitian differential equation

${\displaystyle \ y''-2xy'+2ny=0,\ n\in \mathbb {Z} }$

• Hypergeometric differential equation

${\displaystyle \ x(x-1)y''+\left((\alpha +\beta +1)x-\gamma \right)y'+\alpha \beta y=0,\ \alpha ,\beta ,\gamma \in \mathbb {R} }$

• Laguerre's differential equation

${\displaystyle x\,y''+(1-x)\,y'+ny=0,\ n\in \mathbb {N} _{0}}$.

• Legendre's differential equation

${\displaystyle \ (1-x^{2})y''-2xy'+n(n+1)y=0}$

• Chebyshev differential equation

${\displaystyle \ (1-x^{2})y''-xy'+n^{2}y=0}$

In classical mechanics , the independent variable of the differential equations is often time.

• The differential equation of the harmonic oscillator

${\displaystyle {\ddot {y}}+\omega _{0}^{2}\,y=0}$

Global existence and uniqueness

Be and be arbitrary. Then the initial value problem has a system of linear differential equations ${\displaystyle x_{0}\in I}$ ${\displaystyle y_{0},\dotsc ,y_{n-1}\in \mathbb {R} ^{m}}$

${\displaystyle \left\{{\begin{array}{l}y^{(n)}(x)=\sum _{k=0}^{n-1}a_{k}y^{(k)}(x)+g(x)\ ,\\\ y^{(i)}(x_{0})=y_{i}\ ,\ i=0,\dotsc ,n-1\\\end{array}}\right.}$

according to the global version of Picard-Lindelöf's theorem, exactly one global solution . ${\displaystyle y\colon I\rightarrow \mathbb {R} ^{m}}$

Solution structure

Homogeneous problems

Every linear combination of solutions to a homogeneous problem is in turn a solution - this is called the superposition principle. Thus the set of all solutions is a vector space. In a linear homogeneous differential equation -th order and a linear homogeneous differential equation system of the first order of equations, he is -dimensional. Every basis of the solution space is called a fundamental system . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

Inhomogeneous problems

Knowledge of the fundamental system and a specific solution is sufficient to determine the totality of the solutions to the inhomogeneous problem. Because it is ${\displaystyle y_{sp}\,}$

${\displaystyle \{y=y_{h}+y_{sp}\ |\ y_{h}\ \mathrm {L{\ddot {o}}sung\ des\ homogenen\ Problems} \}}$

the set of all solutions to the inhomogeneous problem.

Special methods for finding a particulate solution

If one has already determined a fundamental system of the associated homogeneous problem, one can construct a special solution of the inhomogeneous problem by the method of variation of the constants or the basic solution procedure described there. If the inhomogeneity shows a particular structure, the exponential approach can occasionally lead to a particular solution more quickly. ${\displaystyle y_{sp}}$

If one has not constructed a fundamental system, a power series approach sometimes works .

Another possibility is the Laplace transformation . The Laplace transform is suitable due to their differentiation theorem, among other things, to solve initial value problems for linear differential equations with constant coefficients. Assuming that the Laplace transform of the inhomogeneity is known, the Laplace transform of the solution is obtained from the differentiation theorem. Under certain circumstances you then know the inverse of it, so that you can recover the (untransformed) solution.

In the special case of a system of differential equations of the first order with constant coefficients, the general solution can be determined with the aid of the matrix exponential function, provided that the Jordanian normal form of the coefficient matrix can be produced.

Periodic Systems

Let be the continuous matrix-valued mapping and the inhomogeneity of the system ${\displaystyle A\colon \mathbb {R} \rightarrow \mathbb {R} ^{m\times m}}$ ${\displaystyle b\colon \mathbb {R} \rightarrow \mathbb {R} ^{m}}$

${\displaystyle y'=A(x)y+b(x)\,.}$

The two maps and are also periodic with the period , that is, and . In general, one cannot explicitly construct a fundamental system of the associated homogeneous problem - but one knows its structure based on Floquet's theorem . ${\displaystyle A}$ ${\displaystyle b}$ ${\displaystyle \omega \in \mathbb {R} }$ ${\displaystyle A(x+\omega )=A(x)}$ ${\displaystyle b(x+\omega )=b(x)}$

With periodic systems the question arises about the existence of periodic solutions with the same period . First one is at the solution space ${\displaystyle \omega }$

${\displaystyle L_{\omega }:=\{y\in C^{1}(\mathbb {R} ;\mathbb {R} ^{m})\ |\ y'(x)=A(x)y(x)\ {\textrm {und}}\ y\ \omega {\textrm {-periodisch}}\}}$

the periodic solutions of the associated homogeneous problem. ${\displaystyle \omega }$

Let be a fundamental matrix of the homogeneous problem . Then the eigenvalues of Floquet multipliers or characteristic multipliers of and are independent of the choice of the fundamental matrix . The following applies: The homogeneous system has a nontrivial -periodic solution if and only if 1 is a Floquet multiplier of . ${\displaystyle \Phi }$ ${\displaystyle y'=A(x)y}$ ${\displaystyle \Phi (\omega )\Phi (0)^{-1}}$ ${\displaystyle y'=A(x)y}$ ${\displaystyle y'=A(x)y}$ ${\displaystyle \omega }$ ${\displaystyle y'=A(x)y}$

For the inhomogeneous problem, one considers the space of the periodic solutions of the adjoint problem ${\displaystyle \omega }$ ${\displaystyle y'=-A(x)^{T}y}$

${\displaystyle L_{\omega }^{\star }:=\{y\in C^{1}(\mathbb {R} ;\mathbb {R} ^{m})\ |\ y'(x)=-A(x)^{T}y(x)\ {\textrm {und}}\ y\ \omega {\textrm {-periodisch}}\}\ .}$

Then the inhomogeneous problem has a -periodic solution if and only if ${\displaystyle y'=A(x)y+b(x)}$ ${\displaystyle \omega }$

${\displaystyle \int _{0}^{\omega }\langle y(s),b(s)\rangle {\rm {d}}s=0}$

applies to all . ${\displaystyle y\in L_{\omega }^{\star }}$

One shows . So for every inhomogeneity has a -periodic solution if 1 is not a Floquet multiplier of . ${\displaystyle \dim L_{\omega }=\dim L_{\omega }^{\star }}$ ${\displaystyle y'=A(x)y+b(x)}$ ${\displaystyle b}$ ${\displaystyle \omega }$ ${\displaystyle y'=A(x)y}$

literature

• Herbert Amann: Ordinary differential equations. 2nd Edition. de Gruyter textbooks, Berlin / New York 1995, ISBN 3-11-014582-0 .
• Carmen Chicone: Ordinary Differential Equations with Applications. 2nd Edition. Texts in Applied Mathematics 34 , Springer-Verlag, 2006, ISBN 0-387-30769-9 .
• Wolfgang Walter : Ordinary differential equations. 3. Edition. Springer Verlag, 1985, ISBN 3-540-16143-0 .