Write a matlab script along with required functions, that solve the following system of differential equations on time horizon [0,1]

y'''(t)+ay(t)=-by''(t)+u(t)

Where u(t) is any function of time

Solution:

% Define the parameters a and b

a = 1;

b = 2;

% Define the time horizon [0,1]

time_horizon = [0, 1];

% Define the initial conditions for y, y', and y''

initials = [0; 0; 0];

% Define the function handle for the input function u(t)

%sin(t) is a common example of a time-varying function.

% You can change the definition of u to any other function of time,

% such as a constant, a step function, or a more complex function, depending on your needs

u = @(t) sin(t);

% Define the function handle for the system of ODEs

odefunction = @(t, y) ode_system(t, y, a, b, u);

% Solve the ODEs using ode45

[t, y] = ode45(odefunction, time_horizon, initials);

% Plot the solution

plot(t, y(:,1), '-', 'LineWidth', 2);

xlabel('t');

ylabel('y');

function dydt = ode_system(t, y, a, b, u)

% Define the system of ODEs

dydt = [y(2); y(3); -b*y(3) + u(t) - a*y(1)];

end

Explain this part in detail dydt = [y(2); y(3); -b*y(3) + u(t) - a*y(1)]; why do we use y(2) y(3) and other similar parts?