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

[Koubaros]

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

 

[da_willem]

Maple returns the following solution :yuck:

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)

 

[dextercioby]

Even though u can make the substitution
$$\frac{dy(x)}{dx} \rightarrow u(x)$$
,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... :tongue2:

EDIT 2:One minute faster,Tide! :tongue2:

 

[Tide]

Koubaros,

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

 

[Koubaros]

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!

 

[saltydog]

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