For the differential equations′′+bs′+5s=0,find the values of b that make the general solution overdamped, underdamped, or critically damped. (for each, give an interval or intervals for b for which the equation is as indicated. thus if the the equation is overdamped for all b in the range −1

overdamped = (-∞,√20) u (√20,∞)

underdamped = (-√20,√20)

critically damped = [√20,√20] u [-√20,-√20]

Step-by-step explanation:

Hi!

In order to answer this question we must first write the charcteristic equation, which is obtained replacing a new variable, lets say ω by the derivatives of s, and putting it to the power of the same order of the derivative, e.g

s''' -> ω^3

The characteristic equation is:

ω^2 + bω + 5 = 0

We can now solve for ω using the quadratic formula:

If b=0, the solution for ω will be a pure complex number, and that means an undamped oscialtor.

The critically damped solution is given when the determinant of ω is zero, that is

b^2 = 20

|b| = √20

The underdamped is given when we have a non-zero imaginary part for ω

that is

b^2<20

|b| < √20

The overdamped is given when the determinant is positive:

b^2>20

|b| > √20

inverse operation of addition