Question

For which value of the constant a>0 does the steady-state solution of x¨+a x˙+4x = 5cos (2t) have maximal amplitude?

(a).a=1
​(b).a = √2 I
(c).a=2
​(d).a=4
(e).there is no such value
​

Answer 1

The value of constant a>0 for maximal amplitude of the steady-state solution of x¸ + a x· + 4x = 5cos(2t) is a=2.

Step-by-step explanation:

The student asks for the value of the constant a>0 that maximizes the amplitude in the steady-state solution of the differential equation x¸ + a x· + 4x = 5cos(2t). This equation represents a dampened harmonic oscillator driven by an external force with angular frequency of 2 rad/s. The amplitude of the resulting steady-state oscillation depends on the damping factor, which is related to the coefficient 'a' in this equation. According to the theory of driven harmonic oscillators, resonance occurs when the driving frequency is equal to the natural frequency of the system, which in this case is √4 rad/s. However, due to damping, the peak amplitude in the real system is achieved at a frequency slightly less than the natural frequency. The condition for maximal amplitude is achieved when the damping coefficient a satisfies the equation a = 2 * √(k/m), where k is the force constant and m is the mass. For an oscillator with a force constant of 4, this simplifies to a = 2, which matches one of the given options.

Answer 2

The value of a that results in maximal amplitude of the steady-state solution for the given differential equation is a = √2. This occurs because at this value, the damped system's resonance frequency matches the driving frequency of the external force, thus satisfying the condition for resonance which maximizes amplitude.

Step-by-step explanation:

The question deals with finding the value of the constant a for which the steady-state solution of the differential equation x´´ + ax´ + 4x = 5cos(2t) has maximal amplitude. This equation describes a damped harmonic oscillator under the influence of an external driving force with a cosine function.

The condition for maximal amplitude in such a system occurs when the driving frequency of the external force matches the natural frequency of the system. This is known as resonance. The equation of motion for a damped harmonic oscillator is of the form m x´´ + b x´ + kx = F0cos(ωt), where m is mass, b is the damping coefficient, k is the spring constant, and F0 is the amplitude of the external force. The natural frequency of the system without damping is given by ω0 = sqrt(k/m). In this case, the system has a natural frequency of ω0 = 2 since the coefficient of the x is 4, implying k/m equals 4, and hence ω0 = sqrt(4).

For maximum amplitude at resonance in a damped system, we use the relationship ωr = sqrt(ω02 - (b/2m)2), where ωr is the resonance frequency. Comparing with the driving frequency of the external force, which is 2, we find that ωr = 2 = sqrt(4 - (a2/4)) where a = b/m. Solving for a, we find that a = sqrt(2). So, the value of the damping coefficient that provides maximal amplitude at steady-state is a = sqrt(2).