The data I have can nicely be described by a power law
$$y=a*t^{-b} + c$$
I need to come up with a differential equation, which the above equation is a solution to. This can be expressed as
$$\frac{dy}{dt}=\frac{-b*y}{t}$$
However, I need to figure out a time-relative expression, i.e. the above is a function of time t and state variable y, whereas I need it only as a function of state variables y[n].
The use case is to have a mechanistical model, showing the behavior under different circumstances. Intuitively, I think it might lead to a space vector of "memory" states or partial derivative, but cant figure out the math.
Thanks for any leads! F/
Not entirely sure of your question but $$y=a*t^{-b} + c$$ becomes after the first derivative $$ \frac{dy}{dt} = -b at^{-b-1} = -\frac{b}{t}(y - c) $$ to get rid of $t$ you can use the fact $$ t^{-b} = \frac{y-c}{a} \to t = \left(\frac{y-c}{a}\right)^{-1/b} $$ or $$ \frac{dy}{dt} = -b at^{-b-1} = -b(y - c)\left(\frac{y-c}{a}\right)^{1/b} = -\frac{b}{a^{1/b}} \left(y-c\right)^{1 + \frac{1}{b}} $$