Please consider the differential equation $$y''+\omega_0^2y=\cos(\omega t)\;,\;\;\;\omega_0,\omega\in(0,\infty)$$ I've actually calculated the general solution $$y(t)=\begin{cases}c_1\cos(\omega_0t)+c_2\sin(\omega_0t)+\frac{1}{\omega_0^2-\omega^2}\cos(\omega t)&\text{, if }\omega_0\ne\omega\\ c_1\cos(\omega t)+\left(c_2+\frac{t}{2\omega}\right)\sin(\omega t)&\text{, otherwise}\end{cases}\;\;\;,\;\;\;\;\;c_1,c_2\in\mathbb{R}$$ Now, I would really like to make a point on how the long term behaviors of these solutions differ. Since the involved trigonometric functions don't converge, it doesn't seem to be sensible to consider $\lim_{t\to\infty}y(t)$.
Obviously, the crucial point is that for $\omega=\omega_0$ resonance occurs and one would expect that the solution diverges to infinity.
For $\omega\not=\omega_0$ (i.e. the system is not at resonance), $y(t)$ is the sinusoid of constant amplitude in your explicit solution, and hence is bounded. The system's response is bounded oscillation of constant amplitude.
On the other hand, when $\omega=\omega_0$ (i.e., the system is at resonance), the lower branch of your explicit solution demonstrates that you get a sinusoidal response with a linearly increasing amplitude, so $\limsup_{t\to\infty}|y(t)|=\infty$.