Week 3 - Solving second order ODEs

%define the value of x
x = 2;

%define an array of values for n (number of terms in the Taylor Series)
n_values = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50];

% Initialize arrays to store n and %Error values
n_array = [];
error_array = [];

%calculate %Error for different values of n
for n = n_values
    %calculate the Taylor Series sum for the given n
    TaylorSeriesSum = 1;  % Initialize with the first term (n = 0)
    factorial = 1;
    power_x = 1;
    
    for i = 1:n
        factorial = factorial * i;
        power_x = power_x * x;
        TaylorSeriesSum = TaylorSeriesSum + power_x / factorial;
    end
    
    %calculate the OriginalValue using the exp(x) function
    OriginalValue = exp(x);
    
    %calculate %Error
    error = ((OriginalValue - TaylorSeriesSum) / OriginalValue) * 100;
    
    % Append n and %Error to the respective arrays
    n_array = [n_array, n];
    error_array = [error_array, error];
end

% Create a plot to visualize the effect of increasing n on %Error
figure;
plot(n_array, error_array, 'k-', 'LineWidth', 1.5, 'DisplayName', 'Error');
title('Effect of Number of Terms on Accuracy of Taylor Series Approximation');
xlabel('Number of Terms (n)');
ylabel('% Error');
legend('Location', 'Northwest');
grid on;