function [x,u1,u2,v1,v2,k,yA, MutlakHata] = nonlineeratis(a,b,alpha,beta,N,M,TOL)

% y'' = f(x,y,y'), y(a)=alpha, y'(a)=t
% z'' = df/dy*z + df/dy'*z' , z(a) = 0, z'(a) = 1

% y = Y0, y' = Y1, df/dy = fY0, df/dy'=fY1

f = @(x,Y0,Y1) ...;
fY0 = @(x,Y0,Y1) ...;
fY1 = @(x,Y0,Y1) ...; 

h = (b-a)/N; % 1/20 = (2-1)/N

x = a:h:b;

t = (beta-alpha)/(b-a);

k = 1;

while 1 
    
    u1(1) = alpha;
u2(1) = t;
v1(1) = 0;
v2(1) = 1;
    
    % ilk denklemin çözümü
    for i = 1:N
        
        k1 = h*u2(i);
        m1 = h*f(x(i),u1(i),u2(i));
        k2 = h*(u2(i) + m1/2);
        m2 = h*f(x(i)+h/2,u1(i)+k1/2,u2(i)+m1/2);
        k3 = h*(u2(i) + m2/2);
        m3 = h*f(x(i)+h/2,u1(i)+k2/2,u2(i)+m2/2);
        k4 = h*(u2(i) + m3);
        m4 = h*f(x(i)+h,u1(i)+k3,u2(i)+m3);
        
        u1(i+1) = u1(i) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);
        u2(i+1) = u2(i) + (1/6)*(m1 + 2*m2 + 2*m3 + m4);
        
    end
    
    if abs(u1(N+1) - beta) < TOL || k>M
        break;
    end
    
    for i = 1:N
        
        kk1 = h*v2(i);
        mm1 = h*(fY0(x(i),u1(i),u2(i))*v1(i) + fY1(x(i),u1(i),u2(i))*v2(i));
        kk2 = h*(v2(i) + mm1/2);
        mm2 = h*(fY0(x(i)+h/2,u1(i),u2(i))*(v1(i)+kk1/2) + fY1(x(i)+h/2,u1(i),u2(i))*(v2(i)+mm1/2));
        kk3 = h*(v2(i) + mm2/2);
        mm3 = h*(fY0(x(i)+h/2,u1(i),u2(i))*(v1(i)+kk2/2) + fY1(x(i)+h/2,u1(i),u2(i))*(v2(i)+mm2/2));
        kk4 = h*(v2(i) + mm3);
        mm4 = h*(fY0(x(i)+h,u1(i),u2(i))*(v1(i)+kk3) + fY1(x(i)+h,u1(i),u2(i))*(v2(i)+mm3));
        
        v1(i+1) = v1(i) + (1/6)*(kk1 + 2*kk2 + 2*kk3 + kk4);
        v2(i+1) = v2(i) + (1/6)*(mm1 + 2*mm2 + 2*mm3 + mm4);
        
    end
    
    %newton metodu
    
    t = t - (u1(N+1) - beta)/v1(N+1);
    
    k = k+1;
    
end

% Analitik çözüm

yA = ...;

MutlakHata = abs(yA - u1);