O: How to solve a 4th order non linear ODE using power series?

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Discussion Overview

The discussion revolves around solving a 4th order ordinary differential equation (ODE) presented by a participant. The equation is initially described as nonlinear, but later clarifications indicate it is linear. Participants explore various methods for finding solutions, including power series and numerical approaches, while expressing concerns about the complexity of the problem and the implications of numerical solutions.

Discussion Character

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

Main Points Raised

  • One participant presents the ODE and requests hints or resources for solving it.
  • Another participant shares a solution obtained from Maple, which involves Bessel and Struve functions, but acknowledges the complexity of the problem.
  • Some participants suggest that a substitution could reduce the order of the ODE, but express that exact solutions may not be feasible due to non-constant coefficients.
  • There is a correction regarding the classification of the ODE from nonlinear to linear, which one participant acknowledges as a mistake.
  • A participant expresses a preference for an elegant analytical solution over numerical methods, citing concerns about symmetry in the equations.
  • Another participant proposes using power series to tackle the problem, noting the singularity at zero but suggesting that a solution may still be attainable.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to solve the ODE. There are competing views on the feasibility of analytical versus numerical solutions, and the discussion reflects uncertainty regarding the implications of the ODE's classification and the methods proposed.

Contextual Notes

The discussion highlights the complexity of the ODE, the potential for singularities, and the challenges associated with finding an elegant solution. There are unresolved mathematical steps and assumptions regarding the methods proposed.

Who May Find This Useful

Individuals interested in solving higher-order ordinary differential equations, particularly in the context of linear equations, power series methods, and numerical analysis may find this discussion relevant.

Koubaros
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Unfortunately, i have ended up with the following 4th order non linear ODE, in a problem i am recently trying to solve. If anyone could give me a hint on how to proceed or point out to me a useful set of notes that i could look into in order to solve it, it would be great.

c*y''''(x)+2*c*(1/x)*y'''(x)-y''(x)-(1/x)*y'(x)=0

c=constant

You can also take a look at the following link for the ODE
http://briefcase.pathfinder.gr/download/o_koubaros/30374/360156/0/ODE.jpg
 
Last edited:
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Maple returns the following solution

y(x) = _C1+_C2*((-1/c)^(1/2)*x*BesselJ(0, (-1/c)^(1/2)*x)+1/2*Pi*(-1/c)^(1/2)*x*(StruveH(0, (-1/c)^(1/2)*x)*BesselJ(1, (-1/c)^(1/2)*x)-StruveH(1, (-1/c)^(1/2)*x)*BesselJ(0, (-1/c)^(1/2)*x)))/(-1/c)^(1/2)+_C3*((-1/c)^(1/2)*x*BesselY(0, (-1/c)^(1/2)*x)+1/2*Pi*(-1/c)^(1/2)*x*(StruveH(0, (-1/c)^(1/2)*x)*BesselY(1, (-1/c)^(1/2)*x)-StruveH(1, (-1/c)^(1/2)*x)*BesselY(0, (-1/c)^(1/2)*x)))/(-1/c)^(1/2)+_C4*Int(x*StruveH(0, (-1/c)^(1/2)*x)*(BesselJ(0, (-1/c)^(1/2)*x)*BesselY(1, (-1/c)^(1/2)*x)-BesselJ(1, (-1/c)^(1/2)*x)*BesselY(0, (-1/c)^(1/2)*x)), x)
 
Even though u can make the substitution
[tex]\frac{dy(x)}{dx} \rightarrow u(x)[/tex]
,which makes it a third order ODE,you cannot solve it exactly,because the coefficients are not constant.Actually the equation is very linear.I can assume u are not too familiar with the classification of ODE-s.Anyway,that's not relevant.
I assume a numerical method might work.Supply initial conditions for the function and its derivatives and a computer software might give you an approximate solution.


Daniel.

EDIT:Maple is a good one...Bessel & Struve... :-p

EDIT 2:One minute faster,Tide! :-p
 
Last edited:
Koubaros,

Incidentally, your DE is NOT nonlinear - it is LINEAR! :-)
 
Sorry

First of all, sorry for the non-linear mistype. Actually it was a "brain" mistype not a "typing" mistype. Anyhow, i was aware of the Maple solution but it does not do any good to me since my problem is rather complex and do need an elegant solution, in order to compare it with solutions from two other domains in my problem.

Unfortunately, it seems like i am going to have to go numerically with this one, which is something that i do not want because it will spoil the symmetry of the equations and of the solution.

Any1 got an idea besides the numericall part, i am all ears!
 
Power series

Hello, has anyone used power series to solve this? Although it's singular at 0, should still be able to get a solution. May look into it but will take time since these are messy and usually need to do a few simple ones before can go to a difficult one.

SD
 

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