But the equation dy/dt = -t*y is not correct by a dimensional point of view.....

You can consider y and t as non-dimensionalised variables.

I would interpret this as a process which gets damped or limited to its immediate vicinity in t or x-domain as the case may be. Something like Subdiffusion

https://en.wikipedia.org/wiki/Anomalous_diffusion

looking at y as a radial direction, that remind me somewhat like a sink at localized action in time, attracting the particle in the origin

Thank you very much for the valuable answers. But may be somebody will propose other idea?

As you correctly deduce, solution of ODE is Gaussian, hence you can apply to any physical system, which have Gaussian profile. For example, Gaussian beam profile in Laser system can be described by your equation. https://en.m.wikipedia.org/wiki/Gaussian_beam

One other case, which comes to mind immediatel, is diffusion equatio, as described by Vikas.

Just a side comment, there is a very interesting equatio, which looks like your equation.

D^2 y=x y

Hi Dr. Džiugys,

If the ODE "dy/dt = − t*y" is rearranged to become "(dy/dt)/t = − y" and t → 0, the process is considered to be undergoing an accelerated exponential decay.

Similar to Newton's law of cooling, you can also find the described dynamical behavior in an accelerated cooling process known as Gaussian Cooling, where the rate of cooling increases with the temperature.

Article Gaussian Cooling and $O^*(n^3)$ Algorithms for Volume and Ga...

Hi Dr. Algis Džiugys

The solution of the Ordinary Differential Equation refers to Gaussian processes, (bacterial mating, radioactive decay, to name a few).

Alan Arturo Flores Cambron explain very well....

As one of the main applications of interest, this equation could describe a linear "damped harmonic oscillator of unit natural frequency and unsteady linear damping".

It has relevance in beam physics and civil engineering, and etc.

what do you means of the *

If * says multiply, then the model solution, e ^ {-1/2(t^2)} + C could stand for an exponential decay.

In continuation, what is internal expectation from dynamical insights after your current equation of concern?

What is its physical model context?

If your your first equation links "attenuation of light", then your current equation should represent "moderated attenuation".

The moderation stems from the new appearance of "position coordinate" on the RHS of your equation.

Thanks to all having participated in this discussion.

إذا كان * يقول multiply ، فإن حل النموذج ، e ^ {-1/2 (t ^ 2)} + C يمكن أن يمثل تحللاً أسيًا.

Elzaki Tarig>

> what do you means of the * ?

"*" stands for multiplication.

P. K. Karmakar:

> What is its physical model context?

System is loosing quantity Y due external influence. Variable y is related to speed of Y change: y = dY/dt. Initially y grows from 0 to max value y_max, then y is stable for some time and finally fall down to 0, because Y goes to 0 too. The equation describes initial stage of the process while Y far from 0.

The equation dy/dt=-t*y describes slightly modified case of the process. More exact equation is dy/dt=-t*(y-y_max).

Thanks to everybody for discussion and very valuable ideas helping to understand why system could follow according to dy/dt=-t*y.

Do you have other ideas?

The general solution of the latter is

S(x)=S(0)exp(-t^2/2). It describes Gaussian distribution.

Degradation of polymer solutions

Dear Vadim,

May be do you have references about that?

The solution of the differential equation y'(t)=-t*y(t) is

y(t)= exp(-0.5*t^2+c)=exp(c)*exp(-0.5*t^2)=k*exp(-0.5*t^2), for t=0 you

get k=y(0).y(t) is a Gauss function in t with a large application region.

Dear Algis, for example,

1) Witold Brostow and Haley E. Hagg Lobland. Lowering mechanical degradation of drag reducers in turbulent flow

2) H.J. Choi, S.T. Lim, Pik-Yin Lai and C.K. Chan. Turbulent Drag Reduction and Degradation of DNA

3) А.Б. Ступин. Кинетическая теория деструкции полимерных молекул в турбулентном потоке