ODE solver with time-dependent term

Do MATLAB ODE solvers usually change the _ * time-dependent terms*_ in the Ordinary differential set of equations? As a simple example, consider the following example from [ 1 ]

function xdot = fun1(t,x,beta,omega,A,w0,theta)
% The time-dependent term is A * sin(w0 * t - theta)
sicw=A * sin(w0 * t - theta);
xdot(2)= -beta*x(2) + omega^2 * x(1) + sicw;
xdot(1) = x(2);
xdot=[xdot(:);sicw]; 
% To make xdot a column
% End of FUN1.M
end

Where calling the function as follow:

beta = .1;
omega = 2;
A = .1;
w0 = 1.3;
theta = pi/4;
X0 = [0 1 0]';
t0 = 0;
tf = 20;
options = [];
[t,y]=ode23(@fun1,[t0,tf],X0,options,beta,omega,A,w0,theta);

When I try to plot the time-dependent term,

sicw=A * sin(w0 * t - theta);

Outside the function or as the third column of the output "y" output versus "t", I get different graphs:

In this example, at least the overall behavior of the graph is preserved, However in my actual code, where the time-dependent term is again a sinusoid,

sicw=0.05e9*sin(2*pi*t/(10e-9))+0.05e9;

the output is worse and even the behavior is not the same!

I don't really know what is going on! Any help is appreciated.

Answer 1

[Jan]

 Outside the function or as the third column of the output "y" output versus "t", I get
 different graphs: ...

Of course you do. Inside the function to be integrated you define get the value:

sicw = A * sin(w0 * t - theta);

and provide it as 3rd component. Then ODE45 integrates this. This must be different from the values defined outside the integration

sicw = A * sin(w0 * t - theta);

because ODE45 replies its integral:

Integral(a * sin(b*t)) = -a/b * cos(b*t) + C

[hossein192003]

Thank you Jan for your answer. In my original code "sicw" is a time-dependent input and I want that to be a sinusoidal function of time, So do I need to define it as:

sicw = (0.05e9*2*pi/10e-9)*cos(2*pi*t/(10e-9));

Where then its integral would be my desired input? Or I do not need to change it and keep it as it is?

Thanks, Hossein

[Jan]

I do not understand the question. Do you want to display sicw or its integral? For the first, you have the formula already, for the second, insert it in the function to be integrated and let ODE45 integrate it. Or use intergral (quad in older Matlab versions).