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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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## Definition of Linear Differential Equations

### Linear nature of equations

Linear differential equations involve the first power of the unknown function and its derivatives, without multiplication between them

### General form of a linear differential equation

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

### Classification of linear differential equations

Linear differential equations can be classified as ordinary (ODEs) or partial (PDEs) depending on the number of independent variables

## Solving Linear Differential Equations

### Methods for solving linear differential equations

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

### Holonomic functions

Solutions to homogeneous linear differential equations with polynomial coefficients, known as holonomic functions, are important for precise calculations and numerical approximations

### Terminology and properties of linear differential equations

The order, nonhomogeneous term, and solution set of a linear differential equation are important concepts in understanding and solving these equations

## Linear Differential Operators

### Definition of linear differential operators

Linear differential operators are combinations of differentiation operators that act on differentiable functions to produce their derivatives

### Properties of linear differential operators

Linear differential operators are linear, form a vector space, and have a kernel that forms a vector space

### Characteristic polynomial and homogeneous equations

The characteristic polynomial is important in solving homogeneous linear differential equations with constant coefficients, and its roots correspond to fundamental solutions of the equation

## Non-Homogeneous Equations and Systems of Linear Differential Equations

### Methods for solving non-homogeneous equations

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 equations and systems of linear differential equations

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 equations and differential Galois theory

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

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