MATLAB: BVP PROBLEM how can we take ‘c’ in x-axis(continuous range.[-1.3:0.02:-1.1]) and to find f”(0)

in this bvp code how can we take 'c' in x-axis(continuous range.[-1.3:0.02:-1.1]) and what is the code to find f"(0)

function main
    S = 1; c  = -1.25; Pr = 0.7; n  = -0.8;
    x  = [3 -1];
    x1 = fsolve(@solver, x);
    function F = solver(x)
        [t, u] = ode45(@equation,[0,5], [S, c, x(1), 1, x(2)]);
             F = [u(end, 2)-1  u(end, 4)];
     figure(1)
    plot(t, u(:,2));
    hold on
    end 
    function dy = equation(t, y)
        dy    = zeros(5,1);
        dy(1) = y(2);
        dy(2) = y(3);
        dy(3) = y(2)^2 - y(1) * y(3) - 1;
        dy(4) = y(5);
        dy(5) = Pr * (n * y(2) * y(4) - y(1) * y(5));
    end
end

Best Answer

function main
    S = 1; c  = [-1.3:0.02:-1.1]; Pr = 0.7; n  = -0.8;
    sol = zeros(numel(c),1);
    x  = [3 -1];
    
    for i = 1:numel(c)
      c_actual = c(i);  
      x1 = fsolve(@solver, x);
      sol(i) = x1(1);
      x = x1;
    end
    plot(c,sol)
    function F = solver(x)
        [t, u] = ode45(@equation,[0,5], [S, c_actual, x(1), 1, x(2)]);
        F = [u(end, 2)-1 , u(end, 4)];
    end 
    function dy = equation(t, y)
        dy    = zeros(5,1);
        dy(1) = y(2);
        dy(2) = y(3);
        dy(3) = y(2)^2 - y(1) * y(3) - 1;
        dy(4) = y(5);
        dy(5) = Pr * (n * y(2) * y(4) - y(1) * y(5));
    end
end
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