Question about second order linear differential equations

In summary, @fresh_42 is asking for help finding a tutorial or article on how to solve a 2nd order Ordinary Differential Equation (ODE) by decomposing it into a series of functions. This can be found in a textbook on differential equations or by searching for "separation of variables" and "ODE decomposition" on Google.
  • #1
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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  • #2
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.)
 
  • #3
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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  • #4
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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  • #5
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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  • #6
see pages 119-125 of these linear algebra notes:

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

1. What is a second order linear differential equation?

A second order linear differential equation is a mathematical equation that involves the derivatives of a function as well as the function itself. It is called "linear" because the function and its derivatives appear in a linear form, meaning there are no products or powers of the function or its derivatives. It is called "second order" because the highest derivative in the equation is of order 2.

2. How do you solve a second order linear differential equation?

The general method for solving a second order linear differential equation involves finding the roots of the characteristic equation, which is obtained by substituting the coefficients of the equation into a general form. The roots then determine the form of the solution, which can be a combination of exponential functions, trigonometric functions, or a mixture of both.

3. What is the difference between a homogeneous and non-homogeneous second order linear differential equation?

A homogeneous second order linear differential equation has a zero term on the right-hand side, meaning that the equation is equal to 0. This type of equation can be solved using the method mentioned above. On the other hand, a non-homogeneous equation has a non-zero term on the right-hand side, which requires the use of additional techniques, such as the method of undetermined coefficients or variation of parameters, to find the particular solution.

4. What is the significance of initial or boundary conditions in solving a second order linear differential equation?

Initial or boundary conditions are necessary in solving a second order linear differential equation because they provide additional information that helps determine the specific solution. These conditions are usually given as the values of the function or its derivatives at a specific point or over a specific interval. Without these conditions, the solution would have an arbitrary constant that cannot be determined.

5. How are second order linear differential equations used in real-world applications?

Second order linear differential equations are used in many fields of science and engineering to model various physical phenomena. For example, they can be used to model the motion of a pendulum, the growth of bacteria, the oscillations in an electrical circuit, and the vibrations of a guitar string. They are also commonly used in the study of mechanics, electromagnetism, and fluid dynamics.

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