The question is in the title, I want to find B and k when they relate to the spring-mass equation that is critically damped. Let me show my work so far.
Here's the spring-mass equation: $$mx''+Bx'+kx=0\;\;\;\;but\;\;\;\;m=1\;\;\;\;x''+Bx'+kx=0$$
Auxillary equation of the above: $$w^2+Bw+k=0$$
Roots: $$w={-B\pm\sqrt{B^2-4(1)k}\over{2(1)}}$$ However, when the equation is critically damped, then $\sqrt{0}$ meaning $B^2=4k$
Anyway, solve for the root to get repeated ones: $$w={{-B\pm0}\over2}={-B\over2},{-B\over2}$$
This is where I am stuck.
My main goal right now is to find $B$ and $k$ for the critically damped system. After that, use them to write the ODE and then solve it. If someone could help me figure this one out it would be greatly appreciated.
Just in case, I have gotten as far as variation of parameters for solving ODEs. Thanks for any help!
$$w={-B\pm\sqrt{B^2-4(1)k}\over{2(1)}}$$ will have double roots if $B^2-4(1)k+0$
You have many choices for $B$ and $K$ for example you may let $k=1$ and $B=2$