Consider the mechanical vibration

$x^{}+2p{x}^{}+{}_0^{2}x=0.$

a$)$ Find the general solution of the ODE using Euler's method.

Hint: Remember what we did in the lecture and how we distinguished several

cases.

b$)$ Reduce this ODE to a $2-$dimensional first$-$order system $x^{}=Ax.$

c$)$ Find the eigenvalues of the system obtained in b$).$ Do you recognize familiari$-$

ties with a$)?$ What can you say about the eigenvectors?