%Question1
clc;clear all; close all;

% Step size
h = 0.1;

% Define the function for the derivative dy/dx
dy_dx = @(x, y) (5 * x^2 - y) / exp(x + y);

% Starting point
x = 0:h:1;
y = zeros(1, length(x));
y(1) = 1;

% Compute derivative and update y using Euler's method
for i = 1:length(x)-1
    F1= h * dy_dx(x(i), y(i));
    F2 = h * dy_dx(x(i) + h/2, y(i) + F1/2);
    F3 = h * dy_dx(x(i) + h/2, y(i) + F2/2);
    F4 = h * dy_dx(x(i) + h, y(i) + F3);
    y(i + 1) = y(i) + (F1 + 2*F2 + 2*F3 + F4) / 6;
end

% Plotting
plot(x, y, 'o-');
grid on
xlabel('x');
ylabel('y');
title('Solution using Fourth-order Runge-Kutta Method');

%-------------------------------------------------------------
%Question2
clc;clear all; close all;

% Step size
h = 0.2;

% Define the function for the derivative dy/dx
dy_dx= @(x, y) (x + y) * sin(x * y);

% Starting point
x = 0:h:2;
y = zeros(1, length(x));
y(1) = 5;

% Compute derivative and update y using fourth-order Runge-Kutta method
for i = 1:length(x)-1
    F1 = h * dy_dx(x(i), y(i));
    F2 = h * dy_dx(x(i) + h/2, y(i) + F1/2);
    F3 = h * dy_dx(x(i) + h/2, y(i) + F2/2);
    F4 = h * dy_dx(x(i) + h, y(i) + F3);
    y(i + 1) = y(i) + (F1 + 2*F2 + 2*F3 + F4) / 6;
end

% Plotting
plot(x, y, 'o-');
grid on;
xlabel('x');
ylabel('y');
title('Solution using Fourth-order Runge-Kutta Method');

----------------------------------------------------------------------
%Question3

clc;
clear all;
close all;

% Step size
h = 0.1;

% Define the function for the derivative dy/dx
dy_dx = @(x, y) (4-y^2)/(2*x);

% Starting point
x = 2:h:3;
y = zeros(1, length(x));
y(1) = 1;
y_euler(1) = 1;

% Compute derivative and update y using Fourth-order Runge-Kutta method
for i = 1:length(x)-1
    F1 = h * dy_dx(x(i), y(i));
    F2 = h * dy_dx(x(i) + h/2, y(i) + F1/2);
    F3 = h * dy_dx(x(i) + h/2, y(i) + F2/2);
    F4 = h * dy_dx(x(i) + h, y(i) + F3);
    y(i + 1) = y(i) + (F1 + 2*F2 + 2*F3 + F4) / 6;
end

% Compute derivative and update y using Euler's method
for i = 1:length(x)-1
    y_euler(i + 1) = y_euler(i) + (h) * dy_dx(x(i), y(i));
end

% Exact solution
f = 2*(3*x.^2-4)./(3*x.^2+4);

% Plotting
plot(x, y, 'oc:', 'LineWidth', 2)
hold on
plot(x, f, 'r--', 'LineWidth', 2);
plot(x, y_euler, 'ob-', 'LineWidth', 2);

xlabel('x');
ylabel('y');
title('Comparison of Fourth-order Runge-Kutta Method and Euler Method');
legend('Runge-Kutta Method', 'Exact Solution', 'Euler Method');
grid on;