Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response All LTI systems possess the eigenfunction property for complex exponential inputs. That is (restricting our attention to periodic complex exponentials), if $e^{j\omega_k t}$ is an input to the LTI system, then the output is $H(j\omega_k) e^{j\omega_k t}$, where $H(j\omega_k)$ is the frequency response of the systsem evaluated at $\omega_k$. When a linear constant coefficient differential equation with zero initial conditions can represent a causal LTI system. Why is it that when we talk about systems, specifically systems with dynamics described by differential equations, we speak of a “steady-state” sinusoidal output? This seems to imply that transients are always present. But if the system is LTI to begin with, would their be transients in the output if a sinusoidal signal, or complex exponential, is applied? It can be shown mathematically that the output of an LTI system when a complex exponential is applied is just a scaled version of the input, with no transient terms. What’s going on here?
It can be shown mathematically that the output of an LTI system when a complex exponential is applied is just a scaled version of the input, with no transient terms.


that's for an input that has been going on since forever ago (and will continue forever to the future):

if
$$ x(t) = e^{j \omega_0 t} \qquad \forall t \in \mathbb{R} $$

then

$$ y(t) = H(j \omega_0) e^{j \omega_0 t} = H(j \omega_0) x(t) $$


  What’s going on here?


problem is, sometimes we have to apply an input at some time we might call "$t=0$", and before the LTI settles down to the $y(t)$ shown above, there are transients.  but if the LTI is stable (and not "marginally stable"), the transients will go away eventually.