Solving Nonlinear ODEs: A Method for Homogeneous and Non-homogeneous Equations"

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Discussion Overview

The discussion revolves around methods for solving nonlinear ordinary differential equations (ODEs), specifically focusing on the transition from homogeneous to non-homogeneous equations. Participants explore potential analytical approaches and the limitations of existing methods, while expressing a desire to avoid numerical solutions when possible.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about methods to solve a non-homogeneous nonlinear ODE given the solution to its homogeneous counterpart, questioning if any analogous techniques exist similar to those for linear ODEs.
  • Another participant notes the inherent difficulty of nonlinear equations and suggests simplifying assumptions, such as linearization or searching for previously solved cases, as potential strategies.
  • A participant emphasizes that the linearity of differential equations allows for separation of solutions, arguing that this principle does not extend to nonlinear equations.
  • Further, a participant discusses the limitations of applying the superposition principle to nonlinear equations, illustrating this with a comparison of linear and nonlinear functions.
  • One participant expresses interest in nonlinear superposition principles and describes a specific case involving a Lane-Emden type ODE, noting that while the homogeneous case is manageable, the addition of a perturbation complicates the solution process.
  • Another participant proposes an Ansatz that combines solutions of the homogeneous and non-homogeneous parts, questioning the validity of this approach and acknowledging the need for approximations.

Areas of Agreement / Disagreement

Participants generally agree on the challenges posed by nonlinear ODEs and the limitations of applying linear methods. However, there is no consensus on a specific method for solving the non-homogeneous case, and multiple competing views on potential approaches remain.

Contextual Notes

Participants mention the need for assumptions and approximations when dealing with nonlinear equations, as well as the potential for unique solutions through educated guessing. The discussion reflects a variety of perspectives on the applicability of known techniques to nonlinear scenarios.

mathphys
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I have a naive question about nonlinear ODEs, supose I have one of such equations, and that it is homogeneous

NL(y(x))=0, (1)

and I know the solution. Then I want to solve the "associated" non-homogeneous equation

NL(y(x))=f(x) (2)

There is a method to attack this kind of problem using the fact that I know the solution of (1)?

I know that for linear ODEs I can use variation of parameters or Green functions, but what about nonlinear ones?

I want to check analytical alternatives before surrender to the temptation of numerical methods.
 
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non-linear equations are quite difficult to handle. there is no straightforward analytic tool.

there are a few options however:

-if the equation you are trying to solve is a model of a physical phenomena, you might try to make some assumptions to simplify it. i.e. you could linearize the equation locally about a point if you are only interested in a certain region
-search to see if your equation has been solved before
-guess. as long as you can satisfy the equation and the bc's your solution should be unique. I mean all diff eq's are in essence solved by guessing, so if you can get an idea of what kind of function will fit the equation, plug it in and give it a go.
 
Remember WHY we could split up our solution procedure for linear diff.eq's by looking at the particular problem and the homogenous problem separately:
If Ly=f(x) is a linear diff. eq., then if Y(x) and Z(x) are two sulitions, then the difference function, Y-Z is a solution of the homogenous problem, due to the linearity of L.

Thus, there is no reason to assume that you can develop something similar for non-linear diff.eqs.
 
The whole point of linearity is that we can split a problem into parts. If y= ax and I replace x with u+ v, then y= a(u+ v)= au+ av, exactly the same as finding y(u) and y(v) separately and then adding them. If y= x2, they y(u+v)= (u+ v)2= u2+ 2uv+ v2= y(u)+ y(v)+ 2uv so knowing y(u) and y(v) does not help me find y(u+v).

If a d.e. is not linear, then knowing a solution to part of the problem doesn't really help!
 
Thanks to you all.
Yep, I understand that the "superposition principle" holds just for linear differential operators. What I'm looking for is a similar (or different) procedure that may be used for nonlinear differential equations. I've heard that there are "nonlinear superposition principles" but I have no idea about how they arise or how I could use them.

I have a lane-emden type ode , which depends on a parameter n.
My equation has a perturbation term, f(x), which depends solely on the independen variable. When n=0,1 the lane-endem ode is a linear homegeneous ode and every thing is nice and easy. When the perturbation is added I still can solve the resulting ode by standard methods.

The point is that I know the solution of the homogenous case for other values of n for which the equation is nonlinear. In this cases the lane emden equation looks like L(y(x))=my(x)^n where L is a linear operator (self-adjoint? , have to check that) and m a constant. It seems like a "generalized" eigen-value problem. But here I have no clue about how to obtain the solution when f(x) is added: L(y(x))=my^n+ f(x)
.
Maybe I will have to guess or to "cook" an approximate solution :bugeye: as some of you suggested. If it doesn't work then i will try numerically :rolleyes:
 
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mm

The simplest way in which I had thought to attack the problem was to assume that the superpostion principle holds anyway, in the following sense:
My Ansatz is

y(x)=y0(x)+y1(x)

where y0 is the solution to

L(y(x))=my(x)^n and ( n not 1 nor 0, sad)


y1 to

L(y(x))=f(x)

This will be a solution, won't it? Maybe not a general one, maybe it won't satisfy some physical restrictions imposed by the problem; but maybe it would work... Is this a valid approach? are there some issues that maybe I've overseen?

Regards

EDIT: Forget this: when y=y0+y1 is replaced on y^n, i would need to do some "suitable" approximations or constraints, of course.
 
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