Determine stability of a scalar linear system given the response?

Question

I am trying to find the stability of a linear scalar system, expressed by: $$ \dot y = y(t)/t + 2u(t) $$ With $ y(t0) = y0 $ and $ t, t0 >0 $

I need to determine if the system is asymptotically stable, and if so, if it is uniformly stable.

I've tried finding the transfer function with the intention of finding the input x(t) expression (and from there the A matrix), but the "1/t" expression cannot be transformed using Laplace (as far as I know).

Any help is appreciated.

Edit: Maybe this is possible using vector norms? Something like: $$ \frac{d}{dt} y(t) = \frac{1}{t}y(t) + 2u(t) $$ $$ \frac{d}{dt} ||y(t)||_2 = ||\frac{1}{t}y(t) + 2u(t)||_2 $$ And then use Lyapunov stability criteria?

Answer 1

I don't remember stability theory any more, but perhaps it is of help that your equation can be solved directly using an integrating factor? Namely, integrating factor can formally be used to note the following idea

$\frac{d}{dt}\frac{y}{t} = \frac{y'}{t} - \frac{y}{t^2} = \frac{1}{t}(y' - \frac{y}{t})$

Thus, substituting the original equation into this result, we get

$\frac{d}{dt}\frac{y}{t} =\frac{2u}{t}$

$d(\frac{y}{t}) =\frac{2u}{t}dt$

$\frac{y}{t} =C + \int_t \frac{2u}{t}dt$

$y(t) = t(C + 2\int_t \frac{u(t)}{t}dt)$

Where $C$ is an arbitrary constant of integration

I suppose you can more easily calculate stability of an explicit expression