91Ó°ÊÓ

When faced with solving a differential equation, a characteristic equation is a tool that simplifies the process. Specifically, for a second-order linear homogeneous differential equation like the one given here, the characteristic equation is derived by substituting a solution of the form \( e^{rt} \) into the differential equation. This approach transforms the differential equation into an algebraic one.

The characteristic equation for the differential equation \( \frac{d^{2}x}{dt^{2}} + 6\frac{dx}{dt} + 9x = 0 \) becomes \( r^2 + 6r + 9 = 0 \). This algebraic equation provides key insights into the behavior of the original differential equation's solutions, depending on the nature of the roots of the characteristic equation.

The differential equation given in the exercise is a classic example of a second-order linear homogeneous differential equation, which generally takes the form \( a\frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + cx = 0 \). In this case, our specific coefficients are a=1, b=6, and c=9.

The 'homogeneous' part of the name indicates that the equation equals zero, and 'second-order' refers to the highest derivative present (in this case, the second derivative of \( x(t) \)). For such equations, solving the characteristic equation yields roots that are essential in determining the general solution. These roots can be real and distinct, a repeating real root, or a pair of complex conjugates.

An initial value problem arises when a differential equation is accompanied by initial conditions. The initial conditions provide the values of the function and possibly its derivatives at a specific point, usually at \( t=0 \).

In this scenario, we were given \( x(0)=0 \) and \( x'(0)=1 \). With the general solution in hand, these initial conditions enable us to determine the specific constants that make the solution unique to the problem at hand—often done by plugging the conditions back into the general solution and its derivatives and solving for the constants.

An overdamped oscillator is a particular type of solution to a second-order linear differential equation, characterized by a lack of oscillatory behavior. Instead of oscillating about the equilibrium position, the system returns to equilibrium in an exponential decay without ever crossing it.

In the given exercise, the presence of a repeated real root in the characteristic equation \( r = -3 \) suggests that the oscillator is overdamped. The displacement \( x(t) = te^{-3t} \) confirms this as it describes a curve that approaches zero from one direction as time progresses, indicative of the overdamped motion you might see in a mechanical system with heavy damping.