"Variation of parameters, also known as variation of constants..."
A technique used to find the general solution to a non-homogeneous linear ODE.
Homogeneous and Non-Homogeneous Differential Equations: Homogeneous and non-homogeneous differential equations are types of differential equations that differ in whether or not the non-homogeneous term is present, where the homogeneous equation has a zero non-homogeneous term.
Linear Differential Equations: Linear differential equations are equations that involve derivatives of an unknown function and are linear in terms of the function and its derivatives.
Method of Undetermined Coefficients: The method of undetermined coefficients is a technique used in mathematics to find a particular solution of a non-homogeneous linear differential equation by assuming a solution form that matches the function on the right-hand side of the equation.
Variation of Parameters: Variation of Parameters is a method used to find a particular solution to a nonhomogeneous linear differential equation by assuming it can be expressed as a linear combination of the solutions to the corresponding homogeneous equation.
Wronskian: The Wronskian is a mathematical tool used to determine the linear independence of a set of solutions of a linear homogeneous differential equation.
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are a fundamental concept in linear algebra that describes specific values and corresponding vectors that remain unchanged when a linear transformation is applied to a vector space.
Laplace Transform: The Laplace Transform is a powerful mathematical tool used to solve differential equations by transforming them from the time domain to the complex frequency domain.
Power Series Method: The power series method is a technique used to solve differential equations by expressing the unknown functions as infinite series.
Boundary Value Problems: Boundary value problems in mathematics refer to a class of differential equations where the values or derivatives of the function are specified at more than one point in the domain.
Partial Differential Equations: Partial Differential Equations (PDEs) are mathematical equations that involve partial derivatives and describe how a quantity changes across multiple dimensions or variables.
"...to solve inhomogeneous linear ordinary differential equations."
"For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients..."
"...those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations."
"Variation of parameters extends to linear partial differential equations as well."
"...linear evolution equations like the heat equation, wave equation, and vibrating plate equation."
"...the method is more often known as Duhamel's principle."
"...named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation."
"...solve the inhomogeneous heat equation."
"Sometimes variation of parameters itself is called Duhamel's principle..."
"...and vice versa."
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"...named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation."
"...linear evolution equations like the heat equation, wave equation, and vibrating plate equation."
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"...to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation."