MATLAB: Boundary condition in non-linear ODE

ode

Dear all,
I wanted to solve this two set of non-linear ODE using matlab :
are constant
The boundary conditions are the following :
and at
and at ( ν is an arbitrary constant < 1)
this the code that I constracted so far 
function bvp4c_mathworks
rspan = [0.01 1];
init = zeros(1,4);
solinit = bvpinit(rspan,init);
sol = bvp4c(@ode4,@bc4,solinit);
eta = sol.x;
theta = sol.y(1,:);
Sr  = sol.y(2,:);
plot(eta,theta)
hold on
plot(eta,Sr,'r')
hold off
legend('Nr(r)','\beta(r)')
end
function du = ode4(eta,u)
theta  = u(1);
Sr   = u(2);             % beta
dtheta = u(3);             % d(theta)/dr
dSr  = u(4);             % d(Sr)/dr
lambda =15.94;
P=12;                    %P=F*a/D; F is the applied force ; a radius of the membrane ; D = E*h^3/12(1-nu^2)
alpha = 3;               %alpha =C*a^2/D ; C in-plane stifnnes
du(1) = dtheta;
du(2) = dSr;
du(3) = (P/(2*pi*eta)-(1/eta)*dtheta+(1/eta^2+lambda^2+Sr));
du(4) = (alpha*theta^2/(2*eta^2)+3/eta*dSr);
du(4) = du(4)/eta;
end
function res = bc4(u0, ur)
res = [ur(1)-0
       ur(2)-0
       ur(3)-0
       u0(2)-0];
end

Best Answer

function du = ode4(eta,u)
  theta  = u(1);
  Sr     = u(2);             % beta
  dtheta = u(3);             % d(theta)/dr
  dSr    = u(4);             % d(Sr)/dr
  lambda = 15.94;
  P=12;                      %P=F*a/D; F is the applied force ; a radius of the membrane ; D = E*h^3/12(1-nu^2)
  alpha = 3;                 %alpha =C*a^2/D ; C in-plane stifnnes
  du = zeros(4,1);
  du(1) = dtheta;
  du(2) = dSr;
  du(3) = P/(2*pi*eta)- 1/eta*dtheta + theta*(1/eta^2+lambda^2+Sr);
  du(4) = -alpha*theta^2/(2*eta^2) - 3/eta*dSr;
end
function res = bc4(ul,ur)
  nu = 0.1;
  res = zeros(4,1);
  res(1) = ul(1);
  res(2) = ur(1);
  res(3) = ul(4);
  res(4) = ur(4)+(1-nu)*ur(2);
end
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