Ordinary differential equations

Variation of parameters

In mathematics, variation of parameters, also known as variation of constants, is a general method 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 with considerably less effort, although 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, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, 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. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. (Wikipedia).

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C29 Variation of parameters Part 2

I continue with an explanation of the method of variation of parameters.

From playlist Differential Equations

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C32 Example problem using variation of parameters

Another example problem using the method of variation of parameters.

From playlist Differential Equations

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C33 Example problem using variation of parameters

Another example problem using the method of variation of parameters on second-order, linear, ordinary DE's.

From playlist Differential Equations

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Variation of parameters

Free ebook http://tinyurl.com/EngMathYT I show how to solve differential equations by applying the method of variation of parameters for those wanting to review their understanding.

From playlist Differential equations

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Differential Equations | Variation of Parameters.

We derive the general form for a solution to a differential equation using variation of parameters. http://www.michael-penn.net

From playlist Differential Equations

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C28 Variation of parameters Part 1

We have already seen variation of parameters in action, but here we expand the method for use in second-order linear DE's, even with non-constant coefficients.

From playlist Differential Equations

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Variation of Constants / Parameters

Download the free PDF http://tinyurl.com/EngMathYT A basic illustration of how to apply the variation of constants / parameters method to solve second order differential equations.

From playlist Differential equations

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Variation of parameters to solve differential equations

Free ebook http://tinyurl.com/EngMathYT How to use the method of variation of parameters to solve second order ordinary differential equations with constant coefficients. Several examples are discussed.

From playlist Differential equations

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C31 The same problem but using variation of parameters

In part 2 I finally solve the example problem using the method of variation of parameters.

From playlist Differential Equations

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From playlist Learning resources

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From playlist StatQuest

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From playlist Summer of Math Exposition Youtube Videos

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From playlist Stanford CS229: Machine Learning Course | Summer 2019 (Anand Avati)

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From playlist Computational Linguistics I

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From playlist DSI Virtual Seminar Series

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From playlist Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series

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A18 Example problem using variation of parameters

An example problem using the method of variation of parameters to solve a non homogeneous system of differential equations.

From playlist A Second Course in Differential Equations

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Variational Methods: How to Derive Inference for New Models (with Xanda Schofield)

This is a single lecture from a course. If you you like the material and want more context (e.g., the lectures that came before), check out the whole course: https://sites.google.com/umd.edu/2021cl1webpage/ (Including homeworks and reading.) Xanda's Webpage: https://www.cs.hmc.edu/~xanda

From playlist Computational Linguistics I

Related pages

Differential operator | Duhamel's principle | Method of undetermined coefficients | Alekseev–Gröbner formula | Characteristic equation (calculus) | Mathematics | Separation of variables | Ordinary differential equation | Wronskian | Reduction of order | Constant of integration | Leonhard Euler | Heat equation | Cramer's rule | Integrating factor | Wave equation | Linear differential equation | Leibniz integral rule