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Hi everyone. I have a project where I need to find a situation this is, or is similar to, a damped oscillator. That is, the Differential Equation (DE) for the system must follow:

x'' + ax' + bx = 0

And, further, it must have some situation corresponding to being 'driven' or 'forced', that is:

x'' + ax' + bx = F

Where F is some term (analogous to a driving force in a mechanical system, which could be a constant, or F=F(x) or F=F(t) or F=F(x,t).

For instance, for a mass on a spring, if x is the displacement, ax' would be the damping term (or a(x')^2), and bx the restoring force. F could be some sinusoidal driving force F(t).

But, I do not want to do the spring on a mass. And not an oscillating electrical circuit either. I though of doing a planet in orbit, moving through some resistive medium, but I cannot think of a driving force. Perhaps another planet coming close to it in its orbit? Would this work? I also thought of a spherical pendulum with two degrees of freedom, but again couldn't think how a driving force comes into it.

Maybe a small ball oscillating (rolling) in a hemi-sphere? If you spin the hemi-sphere does that act as a driving force or not? I think that will only contribute to the damping term, and not correspond to a driving force.

I also thought of doing something from biology. Say, your body's response to a meal. Your blood-sugar levels must decay exponentially right? Or an infection. 'x' could represent the number of bacteria or viruses, then the 'damping' term could be your immune response, and the 'driving force' a drug your taking, but what would the restoring force be (the term proportional to x, bx)?

I am looking for anything that can be modeled by the first DE, and has something corresponding to being driven, that is, the non-homogeneous form,with the "F" on the RHS. It can be from physics, biology, economics, whatever. However, I have only studied physics, and not those other subjects, so it should be something me as an outsider can grasp.

Any help is very much appreciated. Thank you.

x'' + ax' + bx = 0

And, further, it must have some situation corresponding to being 'driven' or 'forced', that is:

x'' + ax' + bx = F

Where F is some term (analogous to a driving force in a mechanical system, which could be a constant, or F=F(x) or F=F(t) or F=F(x,t).

For instance, for a mass on a spring, if x is the displacement, ax' would be the damping term (or a(x')^2), and bx the restoring force. F could be some sinusoidal driving force F(t).

But, I do not want to do the spring on a mass. And not an oscillating electrical circuit either. I though of doing a planet in orbit, moving through some resistive medium, but I cannot think of a driving force. Perhaps another planet coming close to it in its orbit? Would this work? I also thought of a spherical pendulum with two degrees of freedom, but again couldn't think how a driving force comes into it.

Maybe a small ball oscillating (rolling) in a hemi-sphere? If you spin the hemi-sphere does that act as a driving force or not? I think that will only contribute to the damping term, and not correspond to a driving force.

I also thought of doing something from biology. Say, your body's response to a meal. Your blood-sugar levels must decay exponentially right? Or an infection. 'x' could represent the number of bacteria or viruses, then the 'damping' term could be your immune response, and the 'driving force' a drug your taking, but what would the restoring force be (the term proportional to x, bx)?

I am looking for anything that can be modeled by the first DE, and has something corresponding to being driven, that is, the non-homogeneous form,with the "F" on the RHS. It can be from physics, biology, economics, whatever. However, I have only studied physics, and not those other subjects, so it should be something me as an outsider can grasp.

Any help is very much appreciated. Thank you.

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