Proof of angular frequency in damped oscillator [duplicate]

Question

I was given the equation of motion for a damped oscillator is

$$\frac{{\rm d}^2x}{{\rm d}t^2} + \frac{b}{m}\frac{{\rm d}x}{{\rm d}t} + \frac{k}{m}x= 0$$ and the solution of the motion equation is

$$x(t)=A \exp\left(-(b/2m)t\right)\cos(\omega t+\phi).$$ Now how do I go from the solution of the motion equation to this, while using the first and second derivative to solve: $$ω=\sqrt{\frac{k}{m} - \frac{b^2}{4 m^2}}.$$ I feel like I can do it just that I get confused on what is the derivative for A, $\omega$, and $\phi$. Any help will be appreciated thank you.

Answer 1

In general, for an equation of the form $\ddot x + a \dot x + bx = 0$, propose a solution $e^{Rt}$ for $R \in \mathbb C$. Then you will find a quadratic equation for $R$.

Upon solving it, you will find $R = \mu \pm i \nu$ with $\mu,\nu \in \mathbb R$. Using Euler's formula and the fact that linear combinations of solutions are themselves solutions, it can be shown that,

$$x(t) = e^{\mu t}\left( c_1 \cos \nu t + c_2 \sin \nu t\right)$$

assuming no double root ($\nu \neq 0$). You can then apply initial conditions to solve for $c_1$ and $c_2$. This outlines the basic procedure - hope you can take it from here!