In different areas, steady state has slightly different meanings, so please be aware of that.
We want a theory to study the qualitative properties of solutions of differential equations, without solving the equations explicitly.
Moreover, we often want to know whether a certain property of these solutions remains unchanged if the system is subjected to various changes (often called perturbations).
It is very important to be able to study how sensitive the particular model is to small perturbations or changes of initial conditions and of various paramters.
This leads us to an area of DEQ called Stability Analysis using phase space methods and we would consider this for both autonomous and nonautonomous systems under the umbrella of the term equilibrium.
Autonomous
Definition: The equilibrium solution ${y}0$ of an autonomous system $y' = f(y)$ is said to be stable if for each number $\varepsilon$ $>0$ we can find a number $\delta$ $>0$ (depending on $\varepsilon$) such that if $\psi(t)$ is any solution of $y' = f(y)$ having $\Vert$ $\psi(t)$ $- {y_0}$ $\Vert$ $<$ $\delta$, then the solution $\psi(t)$ exists for all $t \geq {t_0}$ and $\Vert$ $\psi(t)$ $- {y_0}$ $\Vert$ $<$ $\varepsilon$ for $t \geq {t_0}$ (where for convenience the norm is the Euclidean distance that makes neighborhoods spherical).
Definition: The equilibrium solution ${y_0}$ is said to be asymptotically stable if it is stable and if there exists a number ${\delta_0}$ $> 0$ such that if $\psi(t)$ is any solution of $y' = f(y)$ having $\Vert$ $\psi(t)$ $- {y_0}$ $\Vert$ $<$ ${\delta_0}$, then $\lim_{t\rightarrow+\infty}$ $\psi(t)$ = ${y_0}$.
The equilibrium solution ${y_0}$ is said to be unstable if it is not stable.
Equivalent definitions can be written for the nonautonomous system $y' = f(t, y)$.
Now we can add notions of globally asymptoctically stable, regions of asymptotic stability and so forth.
From all of these definitions, we can write nice theorems about Linear and Almost Linear system by looking at eigenvalues and we can add notions of conditional stability.
Update
You might also want to peruse the web for notes that deal with the above. For example DEQ.
Regards
The homogeneous form of the solution is actually
$$X_H=c_1e^{-t}sin(5t)+c_2e^{-t}cos(5t)$$
which exponentially decays, so the homogeneous solution is a transient. The steady state solution is the particular solution, which does not decay.
Best Answer
To a differential equation you have two types of solutions to consider: homogeneous and inhomogeneous solutions. The first is the solution to the equation $$x''+2x'+4x=0$$ Taking the tried and true approach of method of characteristics then assuming that $x~e^{rt}$ we have: $$r^2+2r+4=0 \rightarrow (r-r_-)(r-r+)=0 \rightarrow r=r_{\pm}$$ $$r_{\pm}=\frac{-2 \pm \sqrt{4-16}}{2}= -1\pm i \sqrt{3}$$ We see that the homogeneous solution then has the form of decaying periodic functions: $$x_{homogeneous}= Ae^{(-1+ i \sqrt{3})t}+ Be^{(-1- i \sqrt{3})t}=(Ae^{i \sqrt{3}t}+ Be^{- i \sqrt{3}t})e^{-t}$$ Again, these are periodic since we have $e^{i\omega t}$, but they are not steady state solutions as they decay proportional to $e^{-t}$.
The other part of the solution to this equation is then the solution that satisfies the original equation: $$D[x_{inhomogeneous}]= f(t)$$
Upon inspection you can say that this solution must take the form of $Acos(\omega t) + Bsin(\omega t)$. That is because the RHS, f(t), is of the form $sin(\omega t)$. You then need to plug in your expected solution and equate terms in order to determine an appropriate A and B. Once you do this you can then use trig identities to re-write these in terms of c, $\omega$, and $\alpha$.