Solving a Third order ODE

I have been having some trouble solving what I feel like is a simple Third Order ODE in Matlab so I was wondering if there is something obvious that I am doing wrong. My equation is

A'''=((420/(11*Re0))+(g(i)*h(i)+84/11*f(i)*g(i)-35/11*NClam/Re0*h(i)-70/11*Cturb*g(i)*h(i)-70/11*1/Re0*h(i))/f(i);

Where f = A, g = A', h = A'' and the other terms are all constants, Re0 = 28640, NClam = 30*10^6, Cturb = 58. My Boundary Conditions are f(0)=0, g(0)=0, and f(1)=1. I'm using this last boundary condition to shoot for h(0)=?. I have been having some issues because part of the equation is divided by f, so at 0, I can't get a real solution. My code is below, I was hoping someone could show me a way to solve this without getting inf from dividing by 0.

 f=zeros(length(x),1); % f=A
 g=zeros(length(x),1); % g=A'
 h=zeros(length(x),1); % h=A''
 A3=zeros(length(x),1); % A3=A'''
 f(1)=.01; % Boundary Condition at dead end
 g(1)=.001; % Boundary Condition at dead end

 % 1st 2 Shooting Guesses
 SG(1)=.0001;
 SG(2)=.00011;

 for j=1:2
     h(1)=SG(j);
     for i=1:size(x,1)-1
         % Runge Kutta Slopes
         k1f=dx*g(i);
         k1g=dx*h(i);
         k1h=dx*(420/(11*Re0))+(g(i)*h(i)+84/11*f(i)*g(i)-35/11*NClam/Re0*h(i)-70/11*Cturb*g(i)*h(i)-70/11*1/Re0*h(i))/f(i);
         k2f=dx*(g(i)+1/2*k1g);
         k2g=dx*(h(i)+1/2*k1h);
         k2h=dx*(420/(11*Re0))+((g(i)+1/2*k1g)*(h(i)+1/2*k1h)+84/11*(f(i)+1/2*k1f)*(g(i)+1/2*k1g)-35/11*NClam/Re0*(h(i)+1/2*k1h)-70/11*Cturb*(g(i)+1/2*k1g)*(h(i)+1/2*k1h)-70/11*1/Re0*(h(i)+1/2*k1h))/(f(i)+1/2*k1f);
            k3f=dx*(g(i)+1/2*k2g);
            k3g=dx*(h(i)+1/2*k2h);
            k3h=dx*(420/(11*Re0))+((g(i)+1/2*k2g)*(h(i)+1/2*k2h)+84/11*(f(i)+1/2*k2f)*(g(i)+1/2*k2g)-35/11*NClam/Re0*(h(i)+1/2*k2h)-70/11*Cturb*(g(i)+1/2*k2g)*(h(i)+1/2*k2h)-70/11*1/Re0*(h(i)+1/2*k2h))/(f(i)+1/2*k2f);
         k4f=dx*(g(i)+k3g);
         k4g=dx*(h(i)+k3h);
         k4h=dx*(420/(11*Re0))+((g(i)+k3g)*(h(i)+k3h)+84/11*(f(i)+k3f)*(g(i)+k3g)-35/11*NClam/Re0*(h(i)+k3h)-70/11*Cturb*(g(i)+k3g)*(h(i)+k3h)-70/11*1/Re0*(h(i)+k3h))/(f(i)+k3f);

         % Evaluation
         f(i+1)=f(i)+1/6*(k1f+2*k2f+2*k3f+k4f);
         g(i+1)=g(i)+1/6*(k1g+2*k2g+2*k3g+k4g);
         h(i+1)=h(i)+1/6*(k1h+2*k2h+2*k3h+k4h);
     end
     % R is f at channel end, we want equal to Uout
     R(j)=f(size(x,1));
     err=abs(uout-(R(j)));
 end

 % This loop iterates to find the right guess for f''(0) so that
 % f(x/L)= 1

 while err>1e-9
     j=j+1;
     % New guess is found by linear interposation of the last 2 guesses
     SG(j)=SG(j-2)+(SG(j-1)-SG(j-2))/(R(j-1)-R(j-2))*(-R(j-2));
     h(1)=SG(j);
     for i=1:size(x,1)-1
         % Runge Kutta Slopes
         k1f=dx*g(i);
         k1g=dx*h(i);
         k1h=dx*(420/(11*Re0))+(g(i)*h(i)+84/11*f(i)*g(i)-35/11*NClam/Re0*h(i)-70/11*Cturb*g(i)*h(i)-70/11*1/Re0*h(i))/f(i);
         k2f=dx*(g(i)+1/2*k1g);
         k2g=dx*(h(i)+1/2*k1h);
         k2h=dx*(420/(11*Re0))+((g(i)+1/2*k1g)*(h(i)+1/2*k1h)+84/11*(f(i)+1/2*k1f)*(g(i)+1/2*k1g)-35/11*NClam/Re0*(h(i)+1/2*k1h)-70/11*Cturb*(g(i)+1/2*k1g)*(h(i)+1/2*k1h)-70/11*1/Re0*(h(i)+1/2*k1h))/(f(i)+1/2*k1f);
         k3f=dx*(g(i)+1/2*k2g);
         k3g=dx*(h(i)+1/2*k2h);
         k3h=dx*(420/(11*Re0))+((g(i)+1/2*k2g)*(h(i)+1/2*k2h)+84/11*(f(i)+1/2*k2f)*(g(i)+1/2*k2g)-35/11*NClam/Re0*(h(i)+1/2*k2h)-70/11*Cturb*(g(i)+1/2*k2g)*(h(i)+1/2*k2h)-70/11*1/Re0*(h(i)+1/2*k2h))/(f(i)+1/2*k2f);
         k4f=dx*(g(i)+k3g);
         k4g=dx*(h(i)+k3h);
         k4h=dx*(420/(11*Re0))+((g(i)+k3g)*(h(i)+k3h)+84/11*(f(i)+k3f)*(g(i)+k3g)-35/11*NClam/Re0*(h(i)+k3h)-70/11*Cturb*(g(i)+k3g)*(h(i)+k3h)-70/11*1/Re0*(h(i)+k3h))/(f(i)+k3f);

         % Evaluation
         f(i+1)=f(i)+1/6*(k1f+2*k2f+2*k3f+k4f);
         g(i+1)=g(i)+1/6*(k1g+2*k2g+2*k3g+k4g);
         h(i+1)=h(i)+1/6*(k1h+2*k2h+2*k3h+k4h);
     end
     % R is f at channel end, we want equal to Uout
     R(j)=f(size(x,1));
     err=abs(uout-(R(j)));
 end

 % Calculate A'''
 for i = 1:length(x)
     A3(i) = (420/(11*Re0))+(g(i)*h(i)+84/11*f(i)*g(i)-35/11*NClam/Re0*h(i)-70/11*Cturb*g(i)*h(i)-70/11*1/Re0*h(i))/f(i);         end