Explanation Theorem used: The Energy Integral Lemma: Let y ( t ) be a solution to the differential equation y ″ = f ( y ) , where f ( y ) is a continuous function that does not depend on y ′ or the independent variable t . Let F ( y ) be an indefinite integral of f ( y ) . That is, f ( y ) = d d y F ( y ) . Then, the quantity E ( t ) = 1 2 y ′ ( t ) 2 − F ( y ( t ) ) is constant. That is, d d t E ( t ) = 0 . The equation y ″ = f ( y ) can be reduced to 1 2 ( y ′ ) 2 − F ( y ) = K , for some constant K , which is equivalent to the separable first-order equation d y d t = ± 2 [ F ( y ) + K ] having the implicit two-parameter solution t = ± ∫ d y 2 [ F ( y ) + K ] + c . Calculation: The given equation is m y ″ + k y = 0 . Rewrite the given equation as, y ″ = − k m y . Here, f ( y ) = − k m y

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ISBN: 9780135963777

Author: Nagle

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Chapter 4.8, Problem 6E

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To show: The motions of the free undamped mass-spring oscillator my″+ky=0 obey m(y′)2+ky2=constant.

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Chapter 4.8, Problem 5E

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EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

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The motions of the free undamped mass-spring oscillator m y ″ + k y = 0 obey m ( y ′ ) 2 + k y 2 = constant .

Solution Summary: The author explains the Energy Integral Lemma: Let y(t) be a solution to the differential equation.

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To show: The motions of the free undamped mass-spring oscillator my″+ky=0 obey m(y′)2+ky2=constant.

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Chapter 4 Solutions