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Chapter 4: Methods for Linear ODEs of Arbitrary Order
- Linear Equations with Constant Coefficients
- Homogeneous Linear Equations. General Solution
- Nonhomogeneous Linear Equations. General and Particular Solutions
- Linear Equations with Variable Coefficients
- Homogeneous Linear Equations. General Solution. Order Reduction. Liouville Formula
- Nonhomogeneous Linear Equations. General Solution. Superposition Principle
- Nonhomogeneous Linear Equations. Cauchy Problem. Reduction to Integral Equations
- Laplace Transform and the Laplace Integral. Applications to Linear ODEs
- Laplace Transform and the Inverse Laplace Transform
- Main Properties of the Laplace Transform. Inversion Formulas for Some Functions
- Limit Theorems. Representation of Inverse Transforms as Convergent Series and Asymptotic Expansions
- Solution of the Cauchy Problem for Constant-Coefficient Linear ODEs. Applications to Integro-Differential Equations
- Solution of Linear Equations with Polynomial Coefficients Using the Laplace Transform
- Solution of Linear Equations with Polynomial Coefficients Using the Laplace Integral
- Asymptotic Solutions of Linear Equations
- Collocation Method
Abstract
An nth-order homogeneous linear equation with constant coefficients has the general form 4.1.1.1 y x ( n ) + a n - 1 y x ( n - 1 ) + ? + a 1 y x ′ + a 0 y = 0 . . $$ y_{x}^{{(n)}} + a_{{n - 1}} y_{x}^{{(n - 1)}} + \cdots + a_{1} y_{x}^{'} + a_{0} y = 0. . $$
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