Questions: If the differential equation [ m fracd^2 xd t^2+5 fracd xd t+2 x=0 ] is overdamped, the range of values for m is?

$If the differential equation [ m fracd^2 xd t^2+5 fracd xd t+2 x=0 ] is overdamped, the range of values for m is?$

Transcript text: If the differential equation \[ m \frac{d^{2} x}{d t^{2}}+5 \frac{d x}{d t}+2 x=0 \] is overdamped, the range of values for m is? $\square$ Your answer will be an interval of numbers given in the form $(1,2),[1,2),(-i n f, 6]$, etc.

Solution

Solution Steps

Step 1: Identify the Characteristic Equation

The given differential equation is: \[ m \frac{d^{2} x}{d t^{2}} + 5 \frac{d x}{d t} + 2 x = 0 \]

To solve this, we first write the characteristic equation associated with the differential equation. Assume a solution of the form $x(t) = e^{\lambda t}$. Substituting this into the differential equation, we get: \[ m \lambda^2 e^{\lambda t} + 5 \lambda e^{\lambda t} + 2 e^{\lambda t} = 0 \]

Dividing through by $e^{\lambda t}$ (which is never zero), we obtain the characteristic equation: \[ m \lambda^2 + 5 \lambda + 2 = 0 \]

Step 2: Determine the Nature of the Roots

The nature of the roots of the characteristic equation determines whether the system is overdamped, underdamped, or critically damped. The roots are given by the quadratic formula: \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = m$, $b = 5$, and $c = 2$.

Step 3: Condition for Overdamping

For the system to be overdamped, the roots of the characteristic equation must be real and distinct. This occurs when the discriminant is positive: \[ b^2 - 4ac > 0 \]

Substituting the values of $a$, $b$, and $c$: \[ 5^2 - 4 \cdot m \cdot 2 > 0 \] \[ 25 - 8m > 0 \]

Step 4: Solve the Inequality

Solving the inequality for $m$: \[ 25 > 8m \] \[ m < \frac{25}{8} \] \[ m < 3.125 \]