Explanation Given: The value of phase difference γ = 1 L C . Calculation: The impression voltage in L R C -series circuit is E ( t ) = E 0 sin ( γ t ) . The differential equation of the L R C -series circuit is L d 2 q d t 2 + R d q d t + 1 C q = E 0 sin ( γ t ) . The equation of the steady state charge is: q p ( t ) = − E 0 X γ Z 2 sin ( γ t ) − E 0 R γ Z 2 cos ( γ t ) (1) Now, differentiate both sides in equation (1) with respect to t . q ′ p ( t ) = − E 0 X Z 2 cos ( γ t ) + E 0 R Z 2 sin ( γ t ) The above equation is known as the phase current of the L R C -series circuit. i p ( t ) = − E 0 X Z 2 cos ( γ t ) + E 0 R Z 2 sin ( γ t ) Now, the value of the impedance Z = X 2 + R 2 and reactance is X = L γ − 1 C γ . Now, from the properties of the function, y ( x ) = A sin ( x ) + B cos ( x )

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.8, Problem 58E

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The value of the maximum amplitude in LRC− series circuit.

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Chapter 3.8, Problem 57E

Chapter 3.8, Problem 59E

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4. Consider Chebychev's equation (1 - x²)y" - xy + λy = 0 with boundary conditions y(-1) = 0 and y(1) = 0, where X is a constant. (a) Show that Chebychev's equation can be expressed in Sturm-Liouville form d · (py') + qy + Ary = 0, dx y(1) = 0, y(-1) = 0, where p(x) = (1 = x²) 1/2, q(x) = 0 and r(x) = (1 − x²)-1/2 (b) Show that the eigenfunctions of the Sturm-Liouville equation are extremals of the functional A[y], where A[y] = I[y] J[y]' and I[y] and [y] are defined by - I [y] = √, (my² — qy²) dx and J[y] = [[", ry² dx. Explain briefly how to use this to obtain estimates of the smallest eigenvalue >1. 1 (c) Let k > be a parameter. Explain why the functions y(x) = (1-x²) are suitable 4 trial functions for estimating the smallest eigenvalue. Show that the value of A[y] for these trial functions is 4k2 A[y] = = 4k - 1' and use this to estimate the smallest eigenvalue \1. Hint: L₁ x²(1 − ²)³¹ dr = 1 (1 - x²)³ dx (ẞ > 0). 2ẞ

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Chapter 3 Solutions

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The value of the maximum amplitude in L R C − series circuit.

Solution Summary: The author calculates the impression voltage in LRC -series circuit and the differential equation of the steady state charge.

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The value of the maximum amplitude in LRC− series circuit.

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Chapter 3 Solutions