# Difficult ODEs!!

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Posted by Lautaro Vergara on December 16, 1997 at 09:53:25:

Could somebody give some hints about how could I solve
the following stiff ODE?:

(x*y'(x)-3*y(x)=-1/sqrt(2+2*y'+4*x*y'') (1)

with BCs

3*y(0)=1/sqrt(2+2*y'(0))
(2)
y(X)=A*X**3+1/(4*sqrt(30*A)*X)

where X is a suitably chosen 'asymptotic' value
(that could well be 1 or 2) and A is a parameter
that must be numerically found, such that the solution
does not diverge within the interval [0,X].

This is just an example for a more involved set of
coupled nonlinear ODEs of the form

F(y',y,x)=int(G(y'',y',y,x,P),P=1..infinity)

where F is a simple function of their arguments,
like the LHS in (1), the integrals are all finite
(as in the RHS of (1)), G is more complicated but
well behaved function. The BCs that involve a set of
unknown parameters, like A above, that must be
found such that the solutions are finite. The BCs
are such that it there exists at most a discrete
set of acceptable solutions (this is known).

I have tried to implement relaxation, as in
Numerical Recipes, but I have found serious
problems.
I have also converted eq. (1) above as a DAE, but
I have found no code that could sove such system,
when the BCs are NOT consistent (COLDAE does this,
but only for consistent BCs).

I shall appreciate very much some hints about
this or information about a program that could
serve as a basis for solving such kind of problems.

Thank you very much in advance.

Sincerely,

Lautaro Vergara