General Solution to a certain form of ODE

In summary, the conversation discussed a formula that gives a solution to a first order linear ordinary differential equation of the form y' + y = ax^n. The solution involves using an integrating factor and an incomplete gamma function. There does not appear to be a specific name for this type of equation, but there are named equations for first and second order ordinary differential equations in general.
  • #1
CSteiner
31
0
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.
 
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  • #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] .
 
  • #3
yes, that's how I derived it. I was just wondering if it had a name.
 
  • #4
CSteiner said:
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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  • #5


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!
 

1. What is a general solution to a certain form of ODE?

A general solution to a certain form of ODE (ordinary differential equation) is a solution that satisfies the equation for all possible values of the variable. It contains arbitrary constants that can be assigned specific values to obtain a particular solution.

2. How is a general solution to a certain form of ODE different from a particular solution?

A particular solution to a certain form of ODE is a specific solution that satisfies the equation for particular values of the variable. In contrast, a general solution contains arbitrary constants and can be used to find an infinite number of particular solutions.

3. What are the steps to finding a general solution to a certain form of ODE?

The steps to finding a general solution to a certain form of ODE include: identifying the form of the equation, separating variables, integrating both sides, solving for the arbitrary constants, and simplifying the solution.

4. Can a general solution to a certain form of ODE be used to solve any particular problem?

No, a general solution to a certain form of ODE is a more general solution that can be used to solve a family of problems. To solve a specific problem, the arbitrary constants in the general solution need to be assigned specific values based on the initial conditions of the problem.

5. Are there any limitations to finding a general solution to a certain form of ODE?

Yes, there are certain forms of ODEs that do not have a general solution. In these cases, only a particular solution can be found. Additionally, some ODEs may have complex or transcendental solutions that cannot be expressed in terms of elementary functions.

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