Separation of Variables (1st Order) - Revision history

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Johnraymond.yanez: Created page with "* [https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx Separation of Variables Notes]. Produced by Paul Dawkins, Lamar University"

2020-08-12T13:29:03Z

Created page with "* [https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx Separation of Variables Notes]. Produced by Paul Dawkins, Lamar University"

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* [https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx Separation of Variables Notes]. Produced by Paul Dawkins, Lamar University

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 * [https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx Separation of Variables Notes]. Produced by Paul Dawkins, Lamar University
-==References==
+== Licensing ==
-*Polyanin, Andrei D. (2001-11-28). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC. ISBN 1-58488-299-9.
+Content obtained and/or adapted from:
-*Myint-U, Tyn; Debnath, Lokenath (2007). Linear Partial Differential Equations for Scientists and Engineers. Boston, MA: Birkhäuser Boston. doi:10.1007/978-0-8176-4560-1. ISBN 978-0-8176-4393-5.
+* [https://en.wikipedia.org/wiki/Separation_of_variables Separation of variables, Wikipedia] under a CC BY-SA license

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 Hence, the spectral theorem ensures that the separation of variables will (when it is possible) find all the solutions.
-For many differential operators, such as <math>\frac{d^2}{dx^2}</math>, we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).<ref name="Benson2007">David Benson (2007) ''Music: A Mathematical Offering'', Cambridge University Press, Appendix W</ref>
+For many differential operators, such as <math>\frac{d^2}{dx^2}</math>, we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).
 == Matrices ==