BVP with an infinite boundary condition

Hi everybody!

I've written a code that solves the following problem

This is a 2nd order ode and the dependant variable here is rho (tilde) and the independant variable is x (tilde). K (tilde) and B (tilde) are both parameters. The boundary conditions are:

The code includes an optimizer based on the secant method. I'm trying to guess the solution that will ''land'' on the 2nd boundary condition via the shooting method:

function [x2,p2]=optimumup(y21,y22,bc2,l)
%optimizing the choice of y2 for the shooting method, using the Newton
%method (secant) by solving the equation p(length(p(:,1))-bc2=0
[x1,p1] = BP(y21,l);
[x2,p2] = BP(y22,l)
size(p2)
for i=1:10
dp=((p2(length(p2(:,1)),1)-bc2)-(p1(length(p1(:,1)),1)-bc2))/(y22-y21);
y23=y22-(p2(length(p2(:,1)),1)-bc2)/dp;%recurrence relation
p1=p2;
[x2,p2] = BP(y23,l);
if abs(p2(length(p2(:,1)))-bc2)<0.001;
    disp(p2(length(p2(:,1)),1));
    disp(p2(length(p2(:,1)),1)-bc2);
    plot(x2,p2(:,1),x2,p2(:,2))
    return 
end
    y21=p1(1,2);
    y22=p2(1,2);
end
disp('No success')
end


function [x,p] = BP(y2,l)
%Shooting method for the BP eq
options = odeset('RelTol',1e-4,'AbsTol',1e-4);
[x,p] = ode45(@eqs,[0 l],[90/24 y2],options);
%n=size(p);p(n(1),1)
%plot(x,p(:,1),x,p(:,2))
end

function [dy] = eqs(x,y)
%PB equation
dy = zeros(2,1);    % a column vector
kdim=1;Bdim=1;
dy(1) = y(2);
dy(2) = 2.*Bdim./kdim.*y(1)+kdim.^(-1).*log(y(1))-Bdim./kdim;
end

y21 and y22 are the initial x guesses for the secant method. bc2 is the boundary condition at x goes to inifinity. l is the interval size.

I have two main problems:

1. I don't know how to incorperate the inifinite boundary condition.
2. For some reason my solution does not converge. I'm working dimensionless.

I would appreciate some help guys.

Thanks,

Roi