I'm studying a linearization of a differential equation. $x(t)$ and $r(t)$ are really small signals and G, K, B and M are constants. I understand everything until I reach
$$ \frac{d^2x(t)}{dt^2}=G+\frac{K}{M}\sqrt{\left| x(t)-r(t)-\bigg(\frac{MG}{K}\bigg)^2\right|} -\frac{B}{M}\bigg(\frac{dx(t)}{dt}-\frac{dr(t)}{dt}\bigg)$$
And I don't know how they pass from that to:
$$ \frac{d^2x(t)}{dt^2}=-\frac{K^2}{2M^2G} (x(t)-r(t)) -\frac{B}{M}\bigg(\frac{dx(t)}{dt}-\frac{dr(t)}{dt}\bigg)$$
They are obviously performing a linearization of the expression $$G+\frac{K}{M}\sqrt{\left| x(t)-r(t)-\bigg(\frac{MG}{K}\bigg)^2\right|}$$
to
$$\frac{K^2}{2M^2G} (x(t)-r(t)) $$
Can someone clarify what they are doing here? Thanks!
They're using $\sqrt{a^2-b}=a\sqrt{1-b/a^2}\approx a(1- b/(2a^2))$. The last step is the linearization and it is valid when $b/a^2$ is small.
The reason this applies in your situation is you are assuming that x(t)-r(t) is small, so $\sqrt{|x(t)-r(t)- \left(\frac{MG}{K}\right)^2|} = \sqrt{\left(\frac{MG}{K}\right)^2-(x(t)-r(t))}$.