Free oscillations in an over damping spring mass system are governed by the differential equation $ y '' + 5y '+ 6y = 0 $. The mass starts its movement from the starting position $ y (0) = 2 $, with starting velocity $ y '(0) = v_0 $. What is the condition that $ v_0 $ must satisfy for the mass to pass once at equilibrium?
As discussed in the mathematical prelude of chapter 3 in Morin, sometimes the best way to approaching a differential equation is by guessing a general solution of $Ae^{\alpha t}$. Thus we can approach this differential equation with the same approach. Plugging into the given equation, and cancelling the nonzero factor of $Ae^{\alpha t}$, yields $\alpha^2+5\alpha+6=0$. The solutions for $\alpha$ are $\alpha=-2, -3$Call these $\alpha_1$ and $\alpha_2$. Then the general solution to our equation is $$y(t)=Ae^{\alpha_1 t}+Be^{\alpha_2 t}$$ $$y(t)=Ae^{-2t}+Be^{-3t}$$ At $y(0)$ we have $2$, so $2=A+B$ We can differentiate this function for our general equation of $y’(t)$, or $$y’(t)=-2Ae^{-2t}-3Be^{-3t}$$ At $y’(0)$ we have $v_0$, so $$v_0=-2A-3B$$
This is all I have