I was reading this [answer](https://electronics.stackexchange.com/a/368195/236654) in Elecronic engineering stack exchange which said that when solving the linear second order differential equation modelling circuits having ac source. We only need to account for particular solution as homogenous part of the differential always dies to zero.

How do we prove mathematically that for all such linear AC circuits the homogenous part of the solution dies to zero at $t\to +\infty$?

Respone to discussion in comments:One of the has comment on this post that it is direct consequence of solution of differential equations. Ok, so for simplicity I assumed a second order differential equation of form:
$$py''+qy' + r = g(t)$$

Now for the homogenous part I do $$ py'' + gy'+r=0$$ , I find that for the solutions of the above equation to die at infinity, I need both roots of the homogenous part of equation to have have negative real part. Now , I know for sure this isn't always mathematically guaranteed, so why so for linear circuits?