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

$Find the particular solution of the differential equation \frac { w } { g } y ^ { tt } ( t ) + b y ^ { t } ( t ) + k y ( t ) = \frac { w } { g } F ( t ) for the oscillating motion of an object on the end of a spring. In the equation, y is the displacement from equilibrium (positive direction is downward) measured in feet, and t is the time in seconds (see figure). The constant w = 4 is the weight of the object, g = 32 is the acceleration due to gravity, b = \frac { 1 } { 2 } is the magnitude of the resistance to the motion, k = \frac { 25 } { 2 } is the spring constant from Hooke's Law, F ( t ) = 4 \sin 8 t is the acceleration imposed on the system, y ( 0 ) = \frac { 1 } { 5 } and y ^ { t } ( 0 ) = - 6 A) y = \frac { 37 } { 145 } e ^ { - 2 t } \cos ( 4 \sqrt { 6 } t ) + \frac { 217 \sqrt { 6 } } { 870 } e ^ { - 2 t } \sin ( 4 \sqrt { 6 } t ) + \frac { 9 } { 145 } \sin 8 t - \frac { 8 } { 145 } \cos 8 t B) y = \frac { 1 } { 5 } e ^ { 2 t } \sin ( 3 \sqrt { 6 } t ) + \frac { 7 \sqrt { 6 } } { 30 } e ^ { 2 t } \cos ( 3 \sqrt { 6 } t ) + \frac { 7 } { 145 } \sin 8 t + \frac { 8 } { 145 } \cos 8 t C) y = \frac { 37 } { 145 } e ^ { - 2 t } \sin ( 2 \sqrt { 5 } t ) + \frac { 7 \sqrt { 3 } } { 30 } e ^ { - 2 t } \cos ( 2 \sqrt { 5 } t ) - \frac { 9 } { 145 } \sin 8 t - \frac { 7 } { 145 } \cos 8 t D) y = \frac { 1 } { 5 } e ^ { - 2 t } \cos ( 8 \sqrt { 2 } t ) - \frac { 217 \sqrt { 6 } } { 870 } e ^ { - 2 t } \sin ( 8 \sqrt { 2 } t ) + \frac { 8 } { 145 } \sin 8 t - \frac { 9 } { 145 } \cos 8 t E) y = \frac { 1 } { 5 } e ^ { 2 t } \cos ( 7 \sqrt { 3 } t ) - \frac { 7 \sqrt { 3 } } { 30 } e ^ { 2 t } \sin ( 7 \sqrt { 3 } t ) + \frac { 9 } { 145 } \sin 8 t - \frac { 7 } { 145 } \cos 8 t$
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

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

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

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

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

Find the General Solution of the Differential Equation A) 4x2−3xy=C4 x ^ { 2 } - 3 x y = C4x2−3xy=C

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Find the General Solution of the Differential Equation
A) $4 x ^ { 2 } - 3 x y = C$

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