%Question1
clc;clear all; close all;

% Step size
h = 0.1;

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

% Starting point
x = 2:h:3;

y = zeros(1, length(x));
y(1) = exp(1);

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

% Exact solution
f = exp(0.5 * x);

% Plotting
plot(x, y, 'b-', 'LineWidth', 2)
hold on
plot(x, f, 'r--', 'LineWidth', 2);
legend('Approximate Solution', 'Exact Solution');
xlabel('x');
ylabel('y');
title('Approximation vs. Exact Solution');

xlim([2, 3]); 
ylim([0, 5]);



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

% Step size
h = 0.1;

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

% Starting point
x = 0:h:3.2;

y = zeros(1, length(x));
y(1) = 4;

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


% Plotting
plot(x, y, 'b-', 'LineWidth', 2)
xlabel('x');ylabel('y');
title('Approximate Solution');