Hello,
I think you've made a mistake just in transformed form (yy' in lieu of 2yy') in your attachment
You are right, but it is just a typing mistake. I forgot the "2" in typing the equation, but I didn't forget the "2" in the calculs. So the series development is correct.
I have two boundary conditions:
y(0)=finit
y(-infinit)=finit
do you think they're enough to derive a closed form for our solution?
Since it is a first order ODE, only one boundary condition can be settled. If you set two conditions, generally a contradiction will occur.
The bondary condition y(0)=finit sets the yo value appearing in the formulas of the coefficients.
Moreover, a condition such y(-infinit)=finit isn't usable in case of limited series development. The series provides an approximate solution only for not too large values of abs(x), but not for x approaching -infinity or +infinity.
So, don't expect that the solution given in terms of a limited series will be satisfactory in case of large negative x values.
Note:
It seems that the ODE :
yy'-axy=bx^5-cx^3
associated with the boundary condition:
y(-infinit)=finit
has no solution. So, the ODE migth be not convenient to model the physical phenomena which gives y(-infinit)=finit
As a matter of fact, if y(-infinit) is finit, then y'(-infinity)=0
and cx^3 tends to be negigible compared to bx^5. So, the relationship tends to become equivalent to :
-axy=bx^5
With a finit value of y, this is impossible, because -axy isn't equivalent to bx^5, except if (a=0 and b=0), or if (y=0 and b=0).