To verify that a given function is a solution for the damped oscillator, we typically start with the standard equation of motion for a damped harmonic oscillator, which can be expressed as:

Understanding the Damped Oscillator Equation

The equation is generally written as:

m d²x/dt² + b dx/dt + kx = 0

Here, m is the mass, b is the damping coefficient, k is the spring constant, and x is the displacement. The solution to this equation often takes the form of:

x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi)

where:

• A is the amplitude,
• γ = b/(2m) is the damping ratio,
• ω_d = √(k/m - (b/(2m))²) is the damped frequency,
• φ is the phase constant.

Taking Derivatives

To verify that this function is indeed a solution, we need to compute the first and second derivatives of x(t) with respect to time t.

First Derivative

The first derivative of x(t) is:

dx/dt = -Aγ e^{-\gamma t} \cos(\omega_d t + φ) - Aω_d e^{-\gamma t} \sin(\omega_d t + φ)

This expression combines the effects of both the damping term and the oscillatory term. The first part represents the damping effect, while the second part captures the oscillation.

Second Derivative

The second derivative is obtained by differentiating the first derivative:

d²x/dt² = Aγ² e^{-\gamma t} \cos(\omega_d t + φ) + 2Aγω_d e^{-\gamma t} \sin(\omega_d t + φ) - Aω_d² e^{-\gamma t} \cos(\omega_d t + φ)

Now, we can substitute these derivatives back into the original damped oscillator equation.

Substituting into the Equation

Substituting dx/dt and d²x/dt² into the damped oscillator equation:

m(Aγ² e^{-\gamma t} \cos(\omega_d t + φ) + 2Aγω_d e^{-\gamma t} \sin(\omega_d t + φ) - Aω_d² e^{-\gamma t} \cos(\omega_d t + φ)) + b(-Aγ e^{-\gamma t} \cos(\omega_d t + φ) - Aω_d e^{-\gamma t} \sin(\omega_d t + φ)) + k(A e^{-\gamma t} \cos(\omega_d t + φ)) = 0

Grouping Terms

After substituting, we can group terms involving e^{-\gamma t}, cos(ω_d t + φ), and sin(ω_d t + φ). The coefficients of these terms must equal zero for the equation to hold true. This leads to a set of equations that relate m, b, k, γ, and ω_d.

Conclusion

If the relationships derived from the coefficients confirm the original parameters of the damped oscillator, then we have successfully verified that the function is indeed a solution to the damped oscillator equation. This process illustrates how derivatives can be used to validate solutions in differential equations, particularly in physics where such models are prevalent.