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General Solution to a certain form of ODE

  1. Dec 18, 2014 #1
    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.
     
  2. jcsd
  3. Dec 19, 2014 #2
    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] .
     
  4. Dec 19, 2014 #3
    yes, that's how I derived it. I was just wondering if it had a name.
     
  5. Dec 20, 2014 #4

    Astronuc

    User Avatar

    Staff: Mentor

    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
     
    Last edited: Dec 20, 2014
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