Linear differential equation
Linear ordinary differential equations or linear ordinary differential equation systems are an important class of ordinary differential equations.
Definition
Linear ordinary differential equations are differential equations of the form
Where an unknown, defined on an interval real, complex or vector-valued function is introduced, which satisfies the equation presented. It denotes the -th derivative of the unknown function. Is equal to the zero function, it is called a homogeneous, otherwise of an inhomogeneous equation. The function is also called inhomogeneity. It is as well as the coefficient functions a continuous, defined on all function. In the case, the vector- square matrices and the equation is a linear differential equation system for the components of the solution function dar. In the important special case in which the not depend on the equation is referred to as a linear differential equation with constant coefficients.
An important property of linear equations is the superposition principle: Solves the equation with inhomogeneity and inhomogeneity, then solves the linear combination equation with inhomogeneity particular sums and multiples of solutions in the homogeneous case, always back solutions. The reason is that a higher discharge depends in a linear manner from the lower leads.
Examples
- The linear differential equation of first order system of equations
- The linear differential equation of order
- Airy differential equation
- Bessel's differential equation
- Euler's differential equation
- Hermitian differential equation
- Laguerre differential equation
- Legendre differential equation
- Tschebyschowsche differential equation
In classical mechanics, the independent variable of the differential equations is often the time.
- The differential equation of the harmonic oscillator
Global existence and uniqueness
Let and be arbitrary. Then has the initial value problem of a linear system of differential equations
According to the global version of the theorem of Picard - Lindelöf exactly a global solution.
Solution structure
Homogeneous problems
Any linear combination of solutions of a homogeneous problem is again a solution - this is called the principle of superposition. Thus, the set of solutions is a vector space. For a linear homogeneous differential equation of order and a linear homogeneous differential equation of first order system of equations he - dimensionally. Each base of the solution space is called a fundamental system.
Inhomogeneous problems
The knowledge of fundamental system and a particular solution is sufficient to determine the set of solutions of non-homogeneous problem. Indeed, it is
The set of all solutions of the inhomogeneous problem.
Special method for finding a particular solution
If you have already determined a fundamental system of the associated homogeneous problem, we can construct a special solution of the inhomogeneous problem by the method of variation of constants, or the fundamental solution method described there. If the inhomogeneity identifies a particular structure, you can occasionally reach the Exponentialansatz faster to a particular solution.
If you have constructed no fundamental system works occasionally a power series approach.
Another possibility is the Laplace transform. The Laplace transform is due to their differentiation theorem, among other things, to solve initial value problems for linear differential equations with constant coefficients. Provided that one knows the Laplace transform of the non-homogeneity, is obtained from the Differentiation is the Laplace transform of the solution. Under certain circumstances, then you know it is the inverse, so that you can recover the ( untransformed ) solution.
In the special case of a first order system of differential equations with constant coefficients can be the general solution with the aid of the matrix exponential determine if one can establish the Jordan normal form of the coefficient matrix.
Periodic systems
Be the continuous matrix-valued mapping and the inhomogeneity of the system
The two figures and also be periodic with the period, that is, it is and. Although one can in general not explicitly construct the fundamental system of the corresponding homogeneous problem - but you know their structure due to the set of Floquet.
It arises in periodic systems, the question of existence of periodic solutions with the same period. First one is the solution space
The - periodic solutions of the corresponding homogeneous problem interested.
Let be a fundamental matrix of the homogeneous problem. Then the eigenvalues of the Floquet multipliers or characteristic multipliers of hot and are independent of the choice of the fundamental matrix. The following applies: The homogeneous system has a non-trivial iff -periodic solution if 1 is a Floquet multiplier of.
For the inhomogeneous problem we consider the space of periodic solutions of the adjoint problem
Then the inhomogeneous problem has exactly then a periodic solution if
Applies to all.
It shows. So for each inhomogeneity has a periodic solution, if one is not a Floquet multiplier of.