Harmonic oscillator and fourier series

[The Oak]

Hello,
Attached are two problems I can not solve, thanks for the help.

The Attempt at a Solution

For the first question, I understand that I need insert A1coswt+A2sinwt into the homogenous equation , but don't know what's then ..

But I'm pretty much lost on both of em :(

 

[vela]

You should really make the effort to use the template and post the relevant information in the thread itself rather than expecting others to download a file just to read the problem.

Here's the text of the first problem:

Consider a damped driven harmonic oscillator, for which $\beta=\omega_0/4$ and the driving force is given by $F=F_0 \cos \omega t$. Our goal is to find the general solution, which is the sum of the complementary and particular solutions:

(a) Write down the equation of motion.

(b) Write down the complementary (transient, homogenous) solution. (No specific initial conditions are given so your answer will contain two arbitrary coefficients.)

(c) For the particular solution, try the following function: (this is equivalent to equation 3.55 but with a different form)

$$x_p(t) = B_1 \cos \omega t + B_2 \sin \omega t$$

Substitute this trial solution into your equation of motion, rearrange, and equate the coefficients to find expressions for B₁ and B₂.

(d) Now write down the expression for the general solution.
​

Start with part (a). What is the equation of motion for a forced, damped harmonic oscillator?

 

[The Oak]

oops sry I thought it's more convienet in this way

I got the Fourier series question solved.

Ok for a damped harmonic oscillator the equation of motion is

mx^.. + cx^. +kx = F0cos(wt)

is that right?

 

[vela]

It's right, except you don't say what c is.