Growth of a Function I came across an interesting question when studying about externally excited, undamped, one degree of freedom spring-mass system.
The question is: In which manner does the amplitude grow when this system is excited at a frequency equal to the natural frequency of the system?
The amplitude function can be calculated to be:
$Y=1/(1-x^2)$
where, $Y$ is the amplitude
$x$ is the ratio of excitation frequency to the natural frequency of the system.
Here we are interested in the case when $x=1$.
At this point, how does the value of $Y$ approaches infinity ( linearly, exponentially or in which manner)?
 A: The harmonic oscillator without friction and with periodic forcing is governed by:
$$
\begin{cases}
F_0\cos(\Omega\,t) - k\,x(t) = m\,x''(t) \\
x(0) = x_0 \\
x'(0) = v_{x,0}
\end{cases}
$$
that is, by defining the pulsation $\omega := \sqrt{\frac{k}{m}}$ and the static displacement $x_{st} := \frac{F_0}{k}$, we have:
$$
\begin{cases}
x''(t) + \omega^2\,x(t) = x_{st}\,\omega^2\cos(\Omega\,t) \\
x(0) = x_0 \\
x'(0) = v_{x,0}
\end{cases}
$$
which is a simple Cauchy problem whose solution turns out to be:
$$
x(t) = x_0\cos(\omega\,t) + \frac{v_{x,0}}{\omega}\,\sin(\omega\,t) + \frac{x_{st}}{1-\frac{\Omega^2}{\omega^2}}\left[\cos\left(\frac{\Omega}{\omega}\,\omega\,t\right)-\cos\left(\omega\,t\right)\right]
$$
that is, defining in turn the parameters:
$$
A_0 := \sqrt{x_0^2 + \frac{v_{x,0}^2}{\omega^2}}\,,
\quad \quad
\varphi_0 := \text{atan2}\left(-\frac{v_{x,0}}{\omega},\,x_0\right),
\quad \quad
\nu := \frac{\Omega}{\omega}
$$
we have:
$$
x(t) = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}}{1+\nu}\,\frac{\cos(\nu\,\omega\,t)-\cos(\omega\,t)}{1-\nu} \quad \quad \forall \; \nu \ne 1\,.
$$
In particular, by dimensioning the temporal variables with the period $T:=\frac{2\,\pi}{\omega}$ and the spatial variables with $x_{st}$, it's possible to think in dimensionless terms, which is always a good thing.
Having done this, all that remains is to analyze the critical case in which $\nu$ tends to $1$:
$$
\begin{aligned}
\lim_{\nu \to 1} x(t)
& = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}}{2} \lim_{\nu \to 1} \frac{\cos\left(\nu\,\omega\,t\right)-\cos\left(\omega\,t\right)}{1-\nu} \\
& = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}}{2} \lim_{h \to 0} \frac{\cos\left((1-h)\,\omega\,t\right)-\cos\left(\omega\,t\right)}{h} \\
& = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}}{2} \lim_{h \to 0} \frac{\cos(\omega\,t)\cos(h\,\omega\,t) + \sin(\omega\,t)\sin(h\,\omega\,t)-\cos\left(\omega\,t\right)}{h} \\
& = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}}{2}\,\cos(\omega\,t)\, \lim_{h \to 0} \frac{\cos(h\,\omega\,t) - 1}{h} + \frac{x_{st}}{2}\,\sin(\omega\,t)\, \lim_{h \to 0} \frac{\sin(h\,\omega\,t)}{h} \\
& = A_0\cos(\omega\,t + \varphi_0) + \frac{x_{st}\,\omega}{2}\,t\,\sin(\omega\,t) \\
\end{aligned}
$$
which is what would have been obtained by solving the Cauchy problem if $\Omega = \omega$ had been set.

Now, in general, the motion turns out to be the superposition of two oscillatory motions with different pulsations $\Omega \ne \omega$, therefore with different periods, which will also be periodic or not depending on whether the division between the two periods is rational or not.
On the other hand, things get interesting when the two pulsations are close but not identical and the amplitudes are comparable, cases in which the resulting motion is like this:

which reproduces a figure known as beats, or periodic variations of the maximum amplitude of the resulting oscillatory motion. This suggests that in the case in which the pulsation of the forcing is equal to the proper pulsation of the harmonic oscillator ($\Omega = \omega$), there can be catastrophically large amplitude oscillations.
This is exactly what happens and is described analytically by the second solution above valid for $\Omega = \omega$. In particular, setting ourselves in the simple case where $x_0 = 0$ and $v_{x,0} = 0$, we obtain:

and also wanting to investigate what happens in energy terms:
$$
U(t) := \frac{1}{2}\,k\,x(t)^2\,,
\quad \quad
K(t) := \frac{1}{2}\,m\,x'(t)^2\,,
\quad \quad
E(t) := U(t) + K(t)
$$
we have:

From these graphs it's evident that in the case in which $\Omega = \omega$ the amplitude of oscillations grows linearly with respect to time and consequently the energy of the oscillator grows quadratically with respect to time. All this is due to the fact that in these circumstances the motion of the harmonic oscillator is never opposed by that of the forcing, but the latter continues to inject energy!
This physical phenomenon is known as resonance and can also be experienced in everyday life. In particular, when we push a person on a swing, to obtain an ever wider motion we know we have to give the swing some thrusts, even relatively small ones, but always synchronized with his motion.
This is the simplest explanation that can be given to justify the above obtained analytically, but subsequently there are a lot of concrete examples where, unfortunately, this phenomenon has occurred causing in the worst case the destruction of the artifact concerned.
