Express the solution of the given initial value problem as a sum of two oscillations, as in Eq. (8). Equation 8 is: x (t) = C cos(wot - a) + Fo/m w² - w² cos wt, The given equation, with initial conditions is: mx" + kx = Fo cos wt, with w = wo; x(0) = 0, x' (0) = vo (8)
Express the solution of the given initial value problem as a sum of two oscillations, as in Eq. (8). Equation 8 is: x (t) = C cos(wot - a) + Fo/m w² - w² cos wt, The given equation, with initial conditions is: mx" + kx = Fo cos wt, with w = wo; x(0) = 0, x' (0) = vo (8)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Differential Equations in Oscillatory Systems
In this section, we explore the solution of the given initial value problem by expressing it as a sum of two oscillations, following the form presented in Equation (8).
Equation (8) is presented as:
\[ x(t) = C \cos(\omega_0 t - \alpha) + \frac{F_0/m}{\omega_0^2 - \omega^2} \cos \omega t, \tag{8} \]
where:
- \( x(t) \) represents the displacement as a function of time.
- \( C \) and \( \alpha \) are constants determined by initial conditions.
- \( \omega_0 \) is the natural angular frequency of the system.
- \( F_0 \) is the amplitude of the external driving force.
- \( m \) is the mass of the oscillating object.
- \( \omega \) is the angular frequency of the driving force.
The initial value problem, with given initial conditions, is formulated as:
\[ mx'' + kx = F_0 \cos \omega t, \]
with the conditions:
- \( \omega = \omega_0 \)
- \( x(0) = 0 \)
- \( x'(0) = v_0 \)
Here,
- \( x'' \) denotes the second derivative of \( x \) with respect to time \( t \), representing acceleration.
- \( k \) is the spring constant.
In solving this problem, the motion is characterized by two components: the natural oscillation of the system and the forced oscillation due to the external driving force. The complete solution is a superposition of these two oscillatory motions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b9e8ebd-ceba-4fff-be05-4ffc155e08e3%2Fdfbc1aea-e0b0-4543-9f25-37d9a82ab97d%2Fmoyc5xc_processed.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations in Oscillatory Systems
In this section, we explore the solution of the given initial value problem by expressing it as a sum of two oscillations, following the form presented in Equation (8).
Equation (8) is presented as:
\[ x(t) = C \cos(\omega_0 t - \alpha) + \frac{F_0/m}{\omega_0^2 - \omega^2} \cos \omega t, \tag{8} \]
where:
- \( x(t) \) represents the displacement as a function of time.
- \( C \) and \( \alpha \) are constants determined by initial conditions.
- \( \omega_0 \) is the natural angular frequency of the system.
- \( F_0 \) is the amplitude of the external driving force.
- \( m \) is the mass of the oscillating object.
- \( \omega \) is the angular frequency of the driving force.
The initial value problem, with given initial conditions, is formulated as:
\[ mx'' + kx = F_0 \cos \omega t, \]
with the conditions:
- \( \omega = \omega_0 \)
- \( x(0) = 0 \)
- \( x'(0) = v_0 \)
Here,
- \( x'' \) denotes the second derivative of \( x \) with respect to time \( t \), representing acceleration.
- \( k \) is the spring constant.
In solving this problem, the motion is characterized by two components: the natural oscillation of the system and the forced oscillation due to the external driving force. The complete solution is a superposition of these two oscillatory motions.
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