Explanation Formula used: The governing equation of motion is m d 2 y d t 2 + δ d y d t + k y = 0 . The Hooks Law states as: F = k x . Theorem used: “If r 1 and r 2 are the two real and unequal roots of the auxiliary equation a r 2 + b r + c = 0 , then y = c 1 e r 1 x + c 2 e r 2 x is the general solution of a y ″ + b y ′ + c y = 0 ”. The general solution y = y ( x ) to the nonhomogeneous differential equation a y ″ + b y ′ + c y = G ( x ) has the form y = y c + y p , where the complementary solution y c is the general solution to the associated homogeneous equation a y ″ + b y ′ + c y = 0 , and y p is any particular solution to the nonhomogeneous equation a y ″ + b y ′ + c y = G ( x ) ”. Calculation: The weight ( w ) of object is 16-lb stretch a spring with displacement ( x ) 4 ft . From the Hooks Law, F = k x . Where k is spring constant, x is displacement of spring and F is force applied on the spring. Substitute the value of w and x in F = k x . 16 = 4 k k = 4 Now, calculate the mass, w = m g Where, g is acceleration due to gravity and its value in term of British system in earth is 32 ft/sec 2 16 = 32 m m = 1 2 s l u g The instantaneous velocity δ = 4.5 . And the external force is f ( t ) = 4 + e − 2 t Now, the governing equation of motion is: m d 2 y d t 2 + δ d y d t + k y = f ( t ) With initial conditions y ( 0 ) = 2 and y ′ ( 0 ) = 4 ft/s Substitute the value of m , k , f ( t ) = 4 + e − 2 t , and δ in m d 2 y d t 2 + δ d y d t + k y = 0 . 1 2 d 2 y d t 2 + 4.5 d y d t + 4 y = 4 + e − 2 t d 2 y d t 2 + 9 d y d t + 8 y = 8 + 2 e − 2 t The auxiliary equation for the above differential equation is: r 2 y + 9 r y + 8 y = 0 ( r 2 + 9 r + 8 ) y = 0 ( r 2 + 8 r + r + 8 ) y = 0 ( r + 8 ) ( r + 1 ) y = 0 . Splitting the middle term to calculate the value of r . The roots of the above equation are ( r + 8 ) ( r + 1 ) = 0 The value of r = − 8 , − 1 . So, the general solution to the above differential equation is y c ( t ) = c 1 e − 8 t + c 2 e − t To solve the nonhomogeneous solution of the form y p = A + B e − 2 t Substitute the derivative of y p in the given equation d 2 y d t 2 + 9 d y d t + 8 y = 8 + 2 e − 2 t and find that A and B must satisfy the equation. 4 B e − 2 t − 18 B e − 2 t + 8 A + 8 B e − 2 t = 8 + 2 e − 2 t − 6 B e − 2 t + 8 A = 8 + 2 e − 2 t − 3 B e − 2 t + 4 A = 4 + e − 2 t Compare both coefficients of the above equation is as follows: 4 A = 4 A = 1 To calculate the value of B. is as follows: − 3 B e − 2 t = e − 2 t − 3 B = 1 B = − 1 3 Now, substitute the value of A and B in y p = A + B e − 2 t

14th Edition

ISBN: 9780136208013

Author: Hass

Publisher: PEARSON

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Chapter 17 Solutions

THOMAS' CALCULUS EARLY...LL W/MYMATHLAB

Ch. 17.1 - In Exercises 1−30, find the general solution of...Ch. 17.1 - In Exercises 1 – 30, find the general solution of...Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Prob. 9E Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Prob. 35E Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Prob. 38E Ch. 17.1 - Prob. 39E Ch. 17.1 - Prob. 40E Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Prob. 43E Ch. 17.1 - Prob. 44E Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.1 - Prob. 52E Ch. 17.1 - Prob. 53E Ch. 17.1 - Prob. 54E Ch. 17.1 - Prob. 55E Ch. 17.1 - In Exercises 56−60, solve the initial value...Ch. 17.1 - Prob. 57E Ch. 17.1 - Prob. 58E Ch. 17.1 - Prob. 59E Ch. 17.1 - Prob. 60E Ch. 17.1 - Prob. 61E Ch. 17.1 - Prob. 62E Ch. 17.1 - Prob. 63E Ch. 17.1 - Prob. 64E Ch. 17.1 - Prob. 65E Ch. 17.1 - Show that if a, b, and c are positive constants,...Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Solve the equations in Exercises 1−16 by the...Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.2 - Prob. 29E Ch. 17.2 - Prob. 30E Ch. 17.2 - Prob. 31E Ch. 17.2 - Prob. 32E Ch. 17.2 - Prob. 33E Ch. 17.2 - Prob. 34E Ch. 17.2 - Prob. 35E Ch. 17.2 - Prob. 36E Ch. 17.2 - Prob. 37E Ch. 17.2 - Prob. 38E Ch. 17.2 - Prob. 39E Ch. 17.2 - Prob. 40E Ch. 17.2 - Prob. 41E Ch. 17.2 - Prob. 42E Ch. 17.2 - Prob. 43E Ch. 17.2 - Prob. 44E Ch. 17.2 - Prob. 45E Ch. 17.2 - Prob. 46E Ch. 17.2 - Prob. 47E Ch. 17.2 - Prob. 48E Ch. 17.2 - Prob. 49E Ch. 17.2 - Prob. 50E Ch. 17.2 - Prob. 51E Ch. 17.2 - Prob. 52E Ch. 17.2 - In Exercises 53–58, verify that the given function...Ch. 17.2 - In Exercises 53–58, verify that the given function...Ch. 17.2 - Prob. 55E Ch. 17.2 - Prob. 56E Ch. 17.2 - Prob. 57E Ch. 17.2 - Prob. 58E Ch. 17.2 - Prob. 59E Ch. 17.2 - Prob. 60E Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - A 20-lb weight is hung on an 18-in. spring and...Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Suppose L = 10 henrys, R = 10 ohms, C = 1/500...Ch. 17.3 - Prob. 26E Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Prob. 15E Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - Prob. 21E Ch. 17.4 - Prob. 22E Ch. 17.4 - Prob. 23E Ch. 17.4 - Prob. 24E Ch. 17.4 - Prob. 25E Ch. 17.4 - Prob. 26E Ch. 17.4 - Prob. 27E Ch. 17.4 - Prob. 28E Ch. 17.4 - Prob. 29E Ch. 17.4 - Prob. 30E Ch. 17.5 - Prob. 1E Ch. 17.5 - Prob. 2E Ch. 17.5 - Prob. 3E Ch. 17.5 - Prob. 4E Ch. 17.5 - Prob. 5E Ch. 17.5 - Prob. 6E Ch. 17.5 - Prob. 7E Ch. 17.5 - Prob. 8E Ch. 17.5 - Prob. 9E Ch. 17.5 - Prob. 10E Ch. 17.5 - Prob. 11E Ch. 17.5 - Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Prob. 17E Ch. 17.5 - Prob. 18E

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