Find the Particular Solution of the Differential Equation $\frac { w } { g } y ^ { tt } ( t ) + b y ^ { t } ( t ) + k y ( t ) = \frac { w } { g } F ( t )$

question 28

Multiple Choice

Find the particular solution of the differential equation $\frac { w } { g } y ^ { tt } ( t ) + b y ^ { t } ( t ) + k y ( t ) = \frac { w } { g } F ( t )$ for the oscillating motion of an object on the end of a spring. In the equation, $y$ is the displacement from equilibrium (positive direction is downward) measured in feet, and $t$ is the time in seconds (see figure) . The constant $w = 4$ is the weight of the object, $g = 32$ is the acceleration due to gravity, $b = \frac { 1 } { 2 }$ is the magnitude of the resistance to the motion, $k = \frac { 25 } { 2 }$ is the spring constant from Hooke's Law, $F ( t ) = 4 \sin 8 t$ is the acceleration imposed on the system, $y ( 0 ) = \frac { 1 } { 5 }$ and $y ^ { t } ( 0 ) = - 6$

• A) \(y = \frac { 37 } { 145 } e ^ { - 2 t } \cos ( 4 \sqrt { 6 } t ) + \frac { 217 \sqrt { 6 } } { 870 } e ^ { - 2 t } \sin ( 4 \sqrt { 6 } t ) + \frac { 9 } { 145 } \sin 8 t - \frac { 8 } { 145 } \cos 8 t\)
• B) \(y = \frac { 1 } { 5 } e ^ { 2 t } \sin ( 3 \sqrt { 6 } t ) + \frac { 7 \sqrt { 6 } } { 30 } e ^ { 2 t } \cos ( 3 \sqrt { 6 } t ) + \frac { 7 } { 145 } \sin 8 t + \frac { 8 } { 145 } \cos 8 t\)
• C) \(y = \frac { 37 } { 145 } e ^ { - 2 t } \sin ( 2 \sqrt { 5 } t ) + \frac { 7 \sqrt { 3 } } { 30 } e ^ { - 2 t } \cos ( 2 \sqrt { 5 } t ) - \frac { 9 } { 145 } \sin 8 t - \frac { 7 } { 145 } \cos 8 t\)
• D) \(y = \frac { 1 } { 5 } e ^ { - 2 t } \cos ( 8 \sqrt { 2 } t ) - \frac { 217 \sqrt { 6 } } { 870 } e ^ { - 2 t } \sin ( 8 \sqrt { 2 } t ) + \frac { 8 } { 145 } \sin 8 t - \frac { 9 } { 145 } \cos 8 t\)
• E)\(y = \frac { 1 } { 5 } e ^ { 2 t } \cos ( 7 \sqrt { 3 } t ) - \frac { 7 \sqrt { 3 } } { 30 } e ^ { 2 t } \sin ( 7 \sqrt { 3 } t ) + \frac { 9 } { 145 } \sin 8 t - \frac { 7 } { 145 } \cos 8 t\)

Examine the role of perceived control in stress management and its variations across different cultural contexts.

Define psychoneuroimmunology and elaborate on the interconnections between the nervous system, endocrine system, and immune system.

Identify and assess gender differences in response to daily hassles and stress.

Evaluate the health impacts of social isolation versus having diverse social networks.

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