Envelope of x-t graph in Damped harmonic oscillations In our lecture, we learnt that the $x-t$ graph of an underdamped harmonic oscillator is basically a sinusoidal curve with a fixed frequency, bounded by an envelope which is an exponentially decaying curve, like this:
Now, we learned to solve the damped ODE $\ddot x+2\beta \dot x+\omega_0^2x=0$ to get the general non-trivial solution $x(t)=e^{-\beta t}\bigg(c_1\exp(i\omega t)+c_2\exp(-i\omega t)\bigg)$ where $\omega=\sqrt{\omega_0^2-\beta^2}$. The envelope is supposed to be $y=e^{-\beta t}$.
The problem is: I want to prove that all the extremas of $y=\exp(-ax)\sin(bx)$ lie on the envelope $y=\exp(-ax)$ and its mirror image about the x-axis.
So, first, for getting the extremas, $\dot y=0$ which gives us $$\exp(-ax)\bigg(-a\sin bx+b\cos bx\bigg)=0$$ or $$\tan bx =\dfrac ba$$. Now suppose that $x=x_0$ satisfies the last condition. Then how do we show that the point $\bigg(x_0, \exp(-ax_0)\sin(bx_0) \bigg)$ must lie on the curve $y=\pm \exp(-ax)$, or $y^2-\exp(-2ax)=0$?
 A: The extreme points of $y = \exp(-ax) \sin (bx)$ do not actually lie on the curves $y = \pm \exp(-ax)$. Here is a diagram of what's going on, with the extrema of $y = \exp(-ax) \sin (bx)$ in red, and the intersection points in black:

The curve does not touch the envelope exactly at its maximum, but a bit later. We can compute the difference by looking at your derivative calculation. The points where it touches the envelope are $\frac1b(\frac\pi2 + k\pi)$ for $k \in \mathbb Z$. The extrema are $\frac1b (\arctan \frac ba + k \pi)$ for $k \in \mathbb Z$. We will never have $\arctan \frac ba = \frac\pi2$.
But to know that the envelope is an envelope, we do not need the extreme points; all we need to know is that because $$-1 \le \sin (bx) \le 1$$ (with equality infinitely often), we also have $$-\exp(-ax) \le \exp(-ax)\sin (bx) \le \exp(-ax)$$ (with equality infinitely often). In other words, the damped sinusoidal always lies between the two exponential curves, and touches them infinitely many times.
