# Question about second order linear differential equations

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.

fresh_42
Mentor
Here are two first page entries of a Google search:
http://www.math.psu.edu/tseng/class/Math251/Notes-2nd order ODE pt2.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.)

jedishrfu
Mentor
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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Mark44
Mentor
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.

jedishrfu
Mark44
Mentor
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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jedishrfu