function [u1, u2, v1, v2, w1, w2, MH, yA] = LineerAtis(a, b, alpha, beta, N)

% y'' = p(x)y' + q(x)y + r(x), y(a) = alpha, y(b) = beta
% u1 = y1, u2 = y1' , v1 = y2, v2 = y2', w1 = y, w2 = y'

p = @(x) 0;
q = @(x) -1; 
r = @(x) 0; 

% Adım uzunluğu h

h = (b-a)/N;

x = a:h:b; 

% Baslangic kosullari

u1(1) = alpha;
u2(1) = 0; 

v1(1) = 0; 
v2(1) = 1; 

% Runge Kutta Yontemi

for i = 1:N
    
    k11 = h*u2(i);
    k12 = h*(p(x(i))*u2(i) + q(x(i))*u1(i) + r(x(i)));
    
    k21 = h*(u2(i) + k12/2);
    k22 = h*(p(x(i) + h/2)*(u2(i) + k12/2) + q(x(i) + h/2)*(u1(i) + k11/2) + r(x(i) + h/2));
    
    k31 = h*(u2(i) + k22/2);
    k32 = h*(p(x(i) + h/2)*(u2(i) + k22/2) + q(x(i) + h/2)*(u1(i) + k21/2) + r(x(i) + h/2));

    k41 = h*(u2(i) + k32);
    k42 = h*(p(x(i) + h)*(u2(i) + k32) + q(x(i) + h)*(u1(i) + k31) + r(x(i) + h));
    
    u1(i+1) = u1(i) + (1/6)*(k11 + 2*k21 + 2*k31 + k41); % y1
    u2(i+1) = u2(i) + (1/6)*(k12 + 2*k22 + 2*k32 + k42); % y1'
    
end

for i = 1:N
    
    m11 = h*v2(i);
    m12 = h*(p(x(i))*v2(i) + q(x(i))*v1(i));
    
    m21 = h*(v2(i) + m12/2);
    m22 = h*(p(x(i) + h/2)*(v2(i) + m12/2) + q(x(i) + h/2)*(v1(i) + m11/2));
    
    m31 = h*(v2(i) + m22/2);
    m32 = h*(p(x(i) + h/2)*(v2(i) + m22/2) + q(x(i) + h/2)*(v1(i) + m21/2));

    m41 = h*(v2(i) + m32);
    m42 = h*(p(x(i) + h)*(v2(i) + m32) + q(x(i) + h)*(v1(i) + m31));
    
    v1(i+1) = v1(i) + (1/6)*(m11 + 2*m21 + 2*m31 + m41); % y2
    v2(i+1) = v2(i) + (1/6)*(m12 + 2*m22 + 2*m32 + m42); % y2'
    
end


for i = 1:N+1
    
    w1(i) = u1(i) + ((beta - u1(N+1))/v1(N+1))*v1(i); % y
    
    w2(i) = u2(i) + ((beta - u1(N+1))/v1(N+1))*v2(i); % y'
    
end

% Analitik çözüm

yA = cos(x) + (sqrt(2) - 1)*sin(x);
    
% MutlakHata

MH = abs(yA - w1);