General Solution to a certain form of ODE

[CSteiner]

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:

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

 

[the_wolfman]

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

$y= a e^{-x}\int{ e^x x^n dx}+ Ce^{-x}$

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

This will reproduce your solution for the case $C=0$ .

 

[CSteiner]

yes, that's how I derived it. I was just wondering if it had a name.

 

[Astronuc]

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

 

[blue_raver22]

Thank you for sharing your discovery! It is always exciting to come across new solutions in mathematics and science. As you mentioned, this particular formula may have already been discovered and named by other researchers. I suggest doing some further research to see if there is an official name for it. Additionally, it would be helpful to provide more information on the formula, such as its derivation and any limitations or assumptions that were made. This will help other scientists and mathematicians understand and potentially build upon your discovery. Keep exploring and sharing your findings!