Question about second order linear differential equations

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Discussion Overview

The discussion revolves around solving second order linear differential equations, specifically focusing on methods involving series of functions. Participants seek resources and explanations for both homogeneous and non-homogeneous equations.

Discussion Character

  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • One participant requests guidance on solving second order linear differential equations using series of functions, expressing a need for resources that cover non-homogeneous cases.
  • Another participant provides links to external resources, including university notes and Wikipedia, suggesting these as starting points for learning.
  • There is a discussion about the interpretation of "series of functions," with one participant questioning whether it refers to separation of variables, while another clarifies that it likely means a sum of functions.
  • A participant explains the standard approach for homogeneous equations, detailing the use of characteristic equations and providing an example with specific roots and a general solution.
  • Several participants recommend textbooks and additional resources, including a specific textbook by Boyce and DiPrima and linear algebra notes, as helpful for understanding the topic.

Areas of Agreement / Disagreement

Participants generally agree on the need for resources and methods to solve second order linear differential equations, but there is some uncertainty regarding the interpretation of "series of functions" and whether it relates to separation of variables.

Contextual Notes

Some assumptions about the definitions and methods of solving differential equations are not explicitly stated, and there may be unresolved steps in the mathematical explanations provided.

Frank Einstein
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Hi everybody.

I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained.

Thanks for reading.
 
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Here are two first page entries of a Google search:
http://www.math.psu.edu/tseng/class/Math251/Notes-2nd%20order%20ODE%20pt2.pdf (Pennsylvania)
http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx (Texas)
Also a good starting point is Wikipedia and its links there:
https://en.wikipedia.org/wiki/Ordinary_differential_equation
(Depending on the languages you speak or understand, it could be also worth it to change between different languages on Wikipedia as they are not 1:1 translations but individual and thus different descriptions of the same topic.)
 
Are you referring to separation of variables when you say y is composed of series of functions?

If so then separation of variables is actually for a product of functions?

https://en.wikipedia.org/wiki/Separation_of_variables

So in addition to what @fresh_42 has provided may I suggest:

www.mathispower4u.com

It's a website that describes many of these ODE and PDE solution recipes in short videos on each type of problem:

http://www.mathispower4u.com/diff-eq.php

You can view them in any order to bolster your understanding of a given solution recipe.
 
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jedishrfu said:
Are you referring to separation of variables when you say y is composed of series of functions?
I don't think he is, but I'm not certain of that.

By "series of functions" I believe he means a sum of two functions. In a homogeneous 2nd order diff. equation, the standard approach is to assume solutions of the form ##y = e^{rx}##. Finding the roots of the characteristic equation for the diff. equation gets you the values of r.

For example, with the differential equation y'' + 5y' + 4y = 0, the characteristic equation is ##r^2 + 5r + 4 = 0##, the roots of which are r = -4 and r = -1. The general solution would then be ##y = c_1e^{-4t} + c_2e^{-t}##. If initial conditions are given, the constants ##c_1## and ##c_2## can be determined.
 
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Frank Einstein said:
I would be extremely thankfull if someone pointed me to some text in which this is done-explained.
Any textbook on differential equations would have a section on this. A textbook that comes to mind is one by Boyce and DiPrima, but there are lots more.
 
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see pages 119-125 of these linear algebra notes:

http://alpha.math.uga.edu/%7Eroy/laprimexp.pdf
 

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