Explanation Given: The general nonlinear spring equation is given by, d 2 x d t 2 + α x + β x 3 = 0 Here α > 0 and β are parameters, is used to model a variety of physical phenomena Approach: A conservative system has the form given by, d 2 x d t 2 + g ( x ) = 0 Here G ( x ) = ∫ g ( x ) d x corresponds to the potential function. And the energy function is given as, E ( x , v ) = v 2 2 + G ( x ) Here G ( x ) = ∫ g ( x ) d x Calculation: The general nonlinear spring system is, d 2 x d t 2 + α x + β x 3 = 0 Compare with the standard conservative system to get, g ( x ) = α x + β x 3 The potential energy is given as, G ( x ) = ∫ g ( x ) d x Substitute the function g ( x ) as, G ( x ) = ∫ g ( x ) d x = ∫ ( α x + β x 3 ) d x = α x 2 2 + β x 4 4 + C Therefore, the potential function is, G ( x ) = α x 2 2 + β x 4 4 + C … ( 1 ) The energy function is given as, E ( x , v ) = v 2 2 + G ( x ) Substitute the function G ( x ) as, E ( x , v ) = v 2 2 + G ( x ) = v 2 2 + α x 2 2 + β x 4 4 + C Substitute the initial condition E ( 0 , 0 ) = 0 to get,

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Author: Nagle

Publisher: PEARSON

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Chapter 12.4, Problem 17E

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The graph of potential plane and the phase plane diagram.

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Chapter 12.4, Problem 16E

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Chapter 12 Solutions

MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.

Ch. 12.2 - In Problem 16, classify the critical point at the...Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - In Problem 712, find and classify the critical...

Ch. 12.2 - In Problem 712, find and classify the critical...Ch. 12.2 - In Problem 712, find and classify the critical...Ch. 12.2 - Prob. 13E Ch. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Show that when the system x(t)=ax+by+p,...Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Show when the roots of the characteristic equation...Ch. 12.2 - Prob. 27E Ch. 12.3 - In Problems 1 -8, show that the given system is...Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Prob. 5E Ch. 12.3 - Prob. 6E Ch. 12.3 - Prob. 7E Ch. 12.3 - Prob. 8E Ch. 12.3 - In Problems 9 -12, find all the critical points...Ch. 12.3 - Prob. 10E Ch. 12.3 - Prob. 11E Ch. 12.3 - In Problems 9 -12, find all the critical points...Ch. 12.3 - In Problems 13-16, convert the second-order...Ch. 12.3 - In Problems 13-16, convert the second-order...Ch. 12.3 - Prob. 15E Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Prob. 20E Ch. 12.3 - van der Pols Equation. a. Show that van der Pols...Ch. 12.3 - Consider the system dxdt=(+)x+y, dydt=x+(+)y,...Ch. 12.3 - Prob. 23E Ch. 12.3 - Show that coexistence occurs in the competing...Ch. 12.3 - When one of the populations in a competing species...Ch. 12.4 - Prob. 1E Ch. 12.4 - Prob. 2E Ch. 12.4 - Prob. 3E Ch. 12.4 - Prob. 4E Ch. 12.4 - Prob. 5E Ch. 12.4 - Prob. 6E Ch. 12.4 - Prob. 7E Ch. 12.4 - Prob. 8E Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Prob. 12E Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Prob. 15E Ch. 12.4 - Prob. 16E Ch. 12.4 - Prob. 17E Ch. 12.4 - Prob. 18E Ch. 12.4 - Prob. 19E Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 4E Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - In problem 9-14, use Lyapunovs direct method to...Ch. 12.5 - In problem 9-14, use Lyapunovs direct method to...Ch. 12.5 - Prob. 11E Ch. 12.5 - Prob. 12E Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - Prob. 15E Ch. 12.5 - Prob. 16E Ch. 12.5 - Prove that the zero solution for a conservative...Ch. 12.6 - Semistable Limit cycle. For the system...Ch. 12.6 - Prob. 2E Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - In Problems 512, either by hand or using a...Ch. 12.6 - Prob. 6E Ch. 12.6 - Prob. 7E Ch. 12.6 - Prob. 8E Ch. 12.6 - In Problems 5-12, either by hand or using computer...Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 11E Ch. 12.6 - In Problems 5-12, either by hand or using computer...Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - Prob. 15E Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - Prob. 17E Ch. 12.6 - Prob. 18E Ch. 12.6 - Prob. 19E Ch. 12.6 - Prob. 20E Ch. 12.6 - Prob. 21E Ch. 12.6 - Prob. 22E Ch. 12.6 - Prob. 23E Ch. 12.6 - Prob. 24E Ch. 12.6 - Prob. 25E Ch. 12.6 - Prob. 26E Ch. 12.6 - Prob. 27E Ch. 12.6 - Prob. 28E Ch. 12.7 - Prob. 1E Ch. 12.7 - Prob. 2E Ch. 12.7 - Prob. 3E Ch. 12.7 - Prob. 4E Ch. 12.7 - Prob. 5E Ch. 12.7 - Prob. 6E Ch. 12.7 - Prob. 9E Ch. 12.7 - Prob. 10E Ch. 12.7 - Prob. 11E Ch. 12.7 - Prob. 12E Ch. 12.7 - Prob. 13E Ch. 12.7 - Prob. 14E Ch. 12.7 - Prob. 15E Ch. 12.7 - Prob. 16E Ch. 12.7 - Prob. 17E Ch. 12.7 - Prob. 18E Ch. 12.8 - Calculate the Jacobian eigenvalues at the critical...Ch. 12.8 - Prob. 2E Ch. 12.8 - Prob. 3E Ch. 12.8 - Prob. 4E Ch. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - Prob. 2RP Ch. 12.RP - Prob. 3RP Ch. 12.RP - Prob. 4RP Ch. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - Prob. 7RP Ch. 12.RP - In Problems 7 and 8, use the potential plane to...Ch. 12.RP - In Problems 9-12, use Lyapunovs direct method to...Ch. 12.RP - Prob. 10RP Ch. 12.RP - In Problems 9-12, use Lyapunovs direct method to...Ch. 12.RP - Prob. 12RP Ch. 12.RP - Prob. 13RP Ch. 12.RP - In Problem 13 and 14, sketch the phase plane...Ch. 12.RP - In Problems 15 and 16, determine whether the given...Ch. 12.RP - Prob. 16RP Ch. 12.RP - In Problems 17 and 18, determine the stability of...Ch. 12.RP - In Problems 17 and 18, determine the stability of...

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The graph of potential plane and the phase plane diagram.

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Chapter 12 Solutions

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The graph of potential plane and the phase plane diagram.