function [w, Deltax, J, F, yA, MH, k] = NonlineerSonluFarklar(a,b,alpha,beta,N,TOL,M)

% y'' = f(x,y,y'), y(a) = alpha, y(b) = beta
% y = Y0, y'=Y1
% df/dy = fY0, df/dy' = fY1

f = @(x,Y0,Y1) (Y1^2)*(x^(-3)) - 9*(Y0^2)*(x^(-5)) + 4*x;
fY0 = @(x,Y0,Y1) -18*Y0*(x^(-5));
fY1 = @(x,Y0,Y1) 2*Y1*(x^(-3));

h = (b-a)/N;

x = a:h:b;

% Başlangıç w değerleri

for i = 1:N+1
    
    w(i,1) = alpha + ((beta-alpha)/(b-a))*h*(i-1);
    
end

k = 1;

while 1
    
    % Jacobian matrisi
    
    J = zeros(N-1);
    
    for i = 1:N-1
        
        J(i,i) = -2 - (h^2)*fY0(x(i+1),w(i+1),(w(i+2)-w(i))/(2*h));
        
    end
    
    for i = 1:N-2
        
        J(i,i+1) = 1 - (h/2)*fY1(x(i+1),w(i+1),(w(i+2)-w(i))/(2*h));
        J(i+1,i) = 1 + (h/2)*fY1(x(i+2),w(i+2),(w(i+3)-w(i+1))/(2*h));
        
    end
    
    % Sağ taraf vektörü F
    
    for i = 1:N-1
        
        F(i,1) = -w(i) + 2*w(i+1) - w(i+2) + (h^2)*f(x(i+1),w(i+1),(w(i+2)-w(i))/(2*h));
        
    end
    
    % Deltax vektörü
    
    v = J\F;    % N-1 elemanlı bir sütun vektör
    
    if norm(v,2) < TOL || k > M 
        break;
    end
    
    Deltax = zeros(N+1,1); 
    Deltax(2:N,1) = v; 
    
    w = w + Deltax; 
    
    k = k+1; 
    
end

% Analitik çözüm

yA = (x.^3).*log(x); 
    
MH = abs(yA' - w);