%Question1
% First part
clear all;clc;

syms x ;
N = 2;M = 2;

% Define variable `k`
k = 3 + 2 * (x^2);

% Initialize `Z` and `V` arrays
Z = zeros(N, M);
V = zeros(N, 1);

% Loop through to calculate `Z` and `V`
for i = 1:N
    w(i) = x - x^(i + 1);
    
    for j = 1:M
        bn(j) = x - x^(j + 1);
        B = bn(j);

        Z(i, j) = Z(i, j) + int(w(i) * (-1 * diff(diff(B, x), x)), x, 0, 1);
        V(i)=int(k * w(i), x, 0, 1);
    end
end

% Define a predefined `Z` matrix
Z = [
    (1/3),    (1/2);
    (1/2),    (4/5)
];
% Second Part
% Define the right-hand side vector
V = [3/5; 11/12];

% Solve the linear equation system
 I = Z \ V

% Plotting results
x = 0:0.1:1;
fx = 13/10 * (x - x.^2) + 1/3 * (x - x.^3);
fe = 5/3 * x - 3/2 * x.^2 - 1/6 * x.^4;

% Plot with two lines
plot(x, fx, 'o-b', 'LineWidth', 2);
hold on;
plot(x, fe, 'r', 'LineWidth', 2);

grid on;