Question

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Answer 1

We are given a second order differential equation with constant coefficients. To find the general solution of such an equation, first we find the corresponding auxiliary equation and solve it. The roots of the auxiliary equation are then used to write the general solution.

So, if the differential equation is , the corresponding auxiliary equation if found by substituting $m^n$ for the n th derivative.

Hence, the auxiliary equation for the given differential equation is the quadratic equation -----(*)

This can easily be solved using the quadratic formula. ( The solutions of are $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ).

So, the roots of our auxiliary equation are $\frac{-b \pm \sqrt{b^2 - 32}}{2}$ . The term inside the square root is called the discriminant, denoted hereafter by D. This determines the type of solutions we get.

When D < 0, (*) has 2 complex roots, D = 0 gives equal real roots, and D > 0 gives 2 distinct real roots.

Whether the general solution is over, under or critically damped depends on D.

1. Over damped: Over damping occurs when there are distinct real roots to (*) ie. D > 0. This is just
62-32 > 0
or
12 > 32
. This happens when $b > \sqrt{32}= \sqrt{16 \times 2} = 4 \sqrt{2}$ . As there is no upper bound, the value of b can go all the way up to infinity.

If the equation is over damped,  
V2bo
or  
b (4V2, o0)

2. Under damped : Under damping corresponds to 2 complex roots of (*). This happens when .

This is just $b < \sqrt{32}$. We have no lower bound on b for this case, so b ranges from - infinity to .

4V2

If the equation is under damped, $b \in (\infty, 4 \sqrt{2})$ .

3. Critically damped: This case happens when D = 0. Solving this equation gives  $b =4 \sqrt{2}, - 4 \sqrt{2}$.

But, we have another property of the damping coefficients that tells us the value of b at which the solution is critically damped is always between the values of b for which the solution is over damped and the values for which it is under damped. This means we can only take the positive root for b.

Hence, if the equation is critically damped, $b =4 \sqrt{2}$,  $b \in [4 \sqrt{2}, 4 \sqrt{2}]$