# Application of linear constant coefficients ODE of the second order [closed]

I've asked this question in math forum. Apparently this question is not welcomed there. So maybe here I can get a proper response. Consider ODE in the form of $$y''+ay'+by=f(t)$$ where $$a$$ and $$b$$ are constants and $$f(t)$$ is a function of $$t$$. In every book I read there are two applications of DEs of this kind: RLC circuits and mechanical oscillation of an object attached to a spring (with or without damper). Is there any other physical system that can be modeled using this DE? I am not expecting a detailed answer, just a short one containing 1 or 2 examples of related physical systems.

• Crossposted from math.stackexchange.com/q/3032499/11127 Dec 10, 2018 at 13:46
• The answer to this question largely depends on what your criteria are for two applications to be distinct. Is a damped pendulum undergoing small oscillations a "different application" than a damped spring? Because you classified them both under "mechanical oscillations." Yet, at the same time, you did not categorize, for example, an RLC circuit and the motion of a charge in a parabolic potential as both "electromagnetic oscillations," even though they are similar in much the same way. So what is your criteria for two applications being distinct? Dec 10, 2018 at 13:49
• Also, the most general application for this system is: any system whose state can be parametrized by a single parameter, subject to a potential energy that has a local minimum, and where the parameter is driven in some way and whose second derivative depends linearly on its first derivative. Using the loosest possible definition for distinct applications (two systems are distinct under this definition if they have different equations of motion), this is the application for this ODE. Dec 10, 2018 at 13:55

Such differential equations can also occur in Control Theory. You have some Input quantity $$A(t)$$ that can be some physical or chemical quantity and a System that has the Output function $$y(t)$$. However, the System tries to react such that e.g. the Output variable is kept constant as much as possible. This can be realized with some Feedback controls that can be (besides controls proportional to an input):
• react proportional to the time derivative $$y'$$
• react to the time integral over the Output quantity $$\int y dt$$