How to you find the general solution of dr/ds=0.05r?

timberwuf8r 2022-09-30 Answered
How to you find the general solution of d r d s = 0.05 r ?
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Answers (1)

Cooper Gillespie
Answered 2022-10-01 Author has 4 answers
We have:

d r d s = 0.05 r

Which is a First Order linear separable DE. We can simply separate the variables to get

  1 r   d r =   0.05   d s

Then integrating gives:

    ln | r | = 0.05 s + C
| r | = e 0.05 s + C
| r | = e 0.05 s e C

And as e x > 0 x , and putting A = e C we can write the solution as:

r = A e 0.05 s
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For a second order differential equation (many physical systems) in one variable, I know "procedures" to compute the energy. Given
q ( t ) = f ( q ( t ) , q ( t ) ) ,     q ( 0 ) = q 0 ,     q ( 0 ) = v 0 ,
if we're lucky we can read off the related Lagrangian L, introduce p = L q , do a Legendre transform and we got the Hamiltonian function H(q,p) for which d d t H ( q ( t ) , p ( t ) ) = 0 for solutions of the differential equation.
We can be more exact and give all the conditions for Noethers theorem to hold and the result is that along the flow X ( t ) = q ( t ) , p ( t ) in phase space given by a solution with initial conditions X(0), the function H(q,p) always takes the same values. It defines surfaces in phase space indexed by initial conditions X(0).
I wonder how to view this for a priori first order systems
q ( t ) = f ( q ( t ) ) ,     q ( 0 ) = q 0 ,
where I think this must be even simpler. E.g. I thing some functional
q ( t ) q ( t ) an integral over   f ( q ( t ) )   something ,
should exists which will be constant for solutions of the equation, i.e. only depend on q 0 . For each f, the functional dependence of this "energy" on " q 0 " will be different.
However, I can't seem to find a general relation. What's the theory behind this, is there an energy'ish time integral of motion? What is the functional dependence on the intial condition, for a suitable constant of motion for for first order systems. And what would be the interpretation, given that we speak of a situation with only one initial condition?
If it is that case that the system is too restricted so that there is no meaningful geometrical interpretation, then let's think about a system of first order differential equations. This is like the one we generated from phase space, except that it doesn't really come from a second order situation and so the intial conditions aren't really e.g. intial position and velocity. I'm pretty sure there are situation where one considers such a directly generated flow (the equation might be more complicated than exponential flow x ˙ x), but I don't recall any talk about the time-constant of motion in these systems, or how to interpret it.

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