Explanation Given information: The given differential equation is w g y ' ' ( t ) + b y ' ( t ) + k y ( t ) = w g F ( t ) where w = 4 , g = 32 , b = 1 2 , k = 25 2 , F ( t ) = 0 and assume that y ( 0 ) = 1 2 and y ' ( 0 ) = − 4 . Concept Used: If y p is the particular solution of the differential equation y ' ' + a y ' + b y = F ( x ) and y h is the general solution, then y = y h + y p is the general solution of the non homogeneous equation. Calculation: Consider the given differential equation is w g y ' ' ( t ) + b y ' ( t ) + k y ( t ) = w g F ( t ) . The coefficients of the differential terms are, w g = 4 32 = 1 8 b = 1 2 k = 25 2 F ( t ) = 0 Now putting these in the differential equation 1 8 y ' ' ( t ) + 1 2 × y ' ( t ) + 25 2 × y ( t ) = 0 The characteristics equation is 1 8 m 2 + 1 2 m + 25 2 = 0 ⇒ m 2 + 4 m + 100 = 0 . First, we have to find the solution of the characteristics equation m 2 + 4 m + 100 = 0 . m 2 + 4 m + 100 = 0 m = − 4 ± 4 2 − 4 ⋅ 1 ⋅ 100 2 m = − 2 ± i 96 Therefore, y h = C 1 e − 2 t cos 96 t + C 2 e − 2 t sin 96 t Let y p be generalized form of 0

11th Edition

ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 16.3, Problem 34E

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To calculate: The displacement y as a function of time t for the oscillating motion described subject to initial condition. Also graph the displacement function using graphing utility. The given differential equation is wgy''(t)+by'(t)+ky(t)=wgF(t) where w=4,g=32,b=12,k=252,F(t)=0.

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Chapter 16.3, Problem 33E

Chapter 16.3, Problem 35E

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Solution Summary: The author calculates the displacement y as a function of time t for the oscillating motion described subject to initial condition.

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To calculate: The displacement y as a function of time t for the oscillating motion described subject to initial condition. Also graph the displacement function using graphing utility. The given differential equation is wgy''(t)+by'(t)+ky(t)=wgF(t) where w=4,g=32,b=12,k=252,F(t)=0.

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Chapter 16 Solutions