Explanation Given: The mass weight is 4 pounds and the spring constant is 3 lb/ft . It is also given that the external force is f ( t ) = e − t . Calculation: The mass is, m = 4 32 = 1 8 slugs The differential equation is, 1 8 d 2 x d t 2 = − 3 x − d x d t + e − t . So, the differential equation is, 1 8 d 2 x d t 2 + 24 x + d x d t − 8 e − t = 0 (1) The initial conditions are x ( 0 ) = 2 and x ′ ( 0 ) = 0 . Therefore, the auxiliary equation is, m 2 + 8 m + 24 = 0 . The values of m is, m = − 8 ± 8 2 − 4 ( 1 ) ( 24 ) 2 ( 1 ) = − 8 ± 64 − 96 2 = − 8 ± i 32 2 = − 4 ± i 2 2 Thus, the complementary function is, x c = e − 4 t ( c 1 cos 2 2 x + c 2 sin 2 2 x ) . Let, the particular function is x p = A e − t . x p = A e − t x ′ p = − A e − t x ″ p = A e − t Substitute the values of x p , x ′ p and x ″ p in the equation (1)

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Ch. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A force...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...

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The equation of motion.

Solution Summary: The author explains the equation of motion, which is x(t)=e-4t

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