General Solution to a certain form of ODE

  • Thread starter CSteiner
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  • #1
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While fiddling around with some very simple linear ODEs, I "discovered" a formula that gives a solution to ODEs of the form: ##y'+y=ax^n ##.

here it is:

7D%5E%7Bn%7D%5Cleft%20%28%20%5Cfrac%7Bd%5Ei%7D%7Bdx%5Ei%7Dax%5En%20%5Cright%20%29%5Ccos%20i%5Cpi.gif


i'm sure that this was discovered before, but i was just wondering if it had any official name or something.
 

Answers and Replies

  • #2
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You can find the general solution to this equation by using an integrating factor...

[itex] y= a e^{-x}\int{ e^x x^n dx}+ Ce^{-x}[/itex]

The integral can be expressed in terms of the incomplete gamma function.
For positive integer [itex] n [/itex] it simplifies to
[itex] \int{ e^x x^n dx}=e^x\left(x^n - n x^{n-1}+n\left(n-1\right)x^{n-2}\dots -1^n n! \right) [/itex]

This will reproduce your solution for the case [itex] C=0 [/itex] .
 
  • #3
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yes, that's how I derived it. I was just wondering if it had a name.
 
  • #4
Astronuc
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yes, that's how I derived it. I was just wondering if it had a name.
The equation in the OP is a first order linear ordinary differential equation of the general form y' + p(x) y = q(x), with p(x) = 1, q(x) = axn.
http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

http://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.html

There does not appear to be a special name for that particular form. Perhaps there was way back when.

Some named first and second order ordinary differential equations found here:
http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html
 
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