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Solutions of the continuity equation are discussed in a paper C.F.Clement in Series A, Mathematical and Physical Sciences, Vol. 364, No. 1716 (Dec. 12, 1978), pp. 107-119 (link to pdf)

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The first this that comes to my mind is the time dependent logistic equation with production: $$\frac{dN}{dt}=r(t)N(t)\bigg(1-\frac{N(t)}{K(t)}\bigg) + A(t)$$ which arises in population dynamics describing the time evolution of a population (animals, cells, etc). The coefficient $r(t)$ represents the growth rate, while $K(t)$ is the carrying capacity, i.e. ...

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Since you want to find the maximum R, you must differentiate R(ω) with respect to ω (that is dR(ω)/dω ) and then set the derivative equal to zero to find at which value of ω you have a maximum (be careful not to accept negative values or zero). Then insert that value of ω into the response amplitude equation to find your maximum. Hope it helps!

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Physics is about quantified observations of nature, modeled mathematically. The mathematical models are rigorous and self consistent but in order to connect to measurements extra postulates are needed, which define the connection of the mathematical formulae to data. Laws are parts of these postulates. In the same way that axioms are not provable within a ...

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Maybe your are looking for a more physical explanation. Imagine you have a material that consists of identical atoms. The atom cores are not stable and therefore we call the material radioactive. With quantum mechanics you can calculate the probability that a atom core that has not decayed yet will decay. If you do this you will find a constant probability!...

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A scientific law is a statement that concisely states an observation about nature that is true for a wide variety of situations. The important part is that the statement is about observations and experiments; it is not an attempt to explain the phenomena. That's a theory's job. For example, the law of conservation of mass is true for all chemical reactions. ...

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You only need differential equations when you are trying to find movements, and the forces change over time (or position). At any given time, the forces can be written as a normal equation. Differential equations come into play when you look at their changing over time.

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