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Linear differential equations are crucial in mathematics, outlining the relationship between functions and their derivatives. They are characterized by their linear nature and can be ordinary or partial, depending on the variables involved. This overview covers solving techniques, properties, and the computational significance of holonomic functions, which include familiar mathematical expressions like exponentials and trigonometric functions.
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Linear differential equations involve the first power of the unknown function and its derivatives, without multiplication between them
Coefficients and independent variable
The general form of a linear differential equation includes coefficients that are functions of the independent variable
Derivatives
The derivatives in a linear differential equation represent the first, second, and nth derivatives of the unknown function with respect to the independent variable
Linear differential equations can be classified as ordinary (ODEs) or partial (PDEs) depending on the number of independent variables
Different methods, such as integration techniques and specialized algorithms, can be used to solve linear differential equations depending on the characteristics of the coefficients and the order of the equation
Solutions to homogeneous linear differential equations with polynomial coefficients, known as holonomic functions, are important for precise calculations and numerical approximations
The order, nonhomogeneous term, and solution set of a linear differential equation are important concepts in understanding and solving these equations
Linear differential operators are combinations of differentiation operators that act on differentiable functions to produce their derivatives
Linear differential operators are linear, form a vector space, and have a kernel that forms a vector space
The characteristic polynomial is important in solving homogeneous linear differential equations with constant coefficients, and its roots correspond to fundamental solutions of the equation
Different methods, such as the method of undetermined coefficients and the method of variation of constants, can be used to solve non-homogeneous linear differential equations with constant coefficients
First-order linear differential equations can be solved by integrating both sides of the equation, while systems of linear differential equations can be solved using matrix operations
Higher-order linear differential equations with variable coefficients are generally not solvable by quadrature, and differential Galois theory provides methods for solving them when possible