Solved: Electric Circuits. 32 and 33 are concerned with the electric circuit described

Chapter 7, Problem 33

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Electric Circuits. 32 and 33 are concerned with the electric circuit described by thesystem of differential equations in of Section 7.1:ddt IV=R1L 1L1C 1CR2IV. (i) Consider the preceding system of differential equations (i).(a) Find a condition on R1, R2, C, and L that must be satisfied if the eigenvalues of thecoefficient matrix are to be real and different.(b) If the condition found in part (a) is satisfied, show that both eigenvalues are negative.Then show that I(t) 0 and V(t) 0 as t , regardless of the initial conditions.(c) If the condition found in part (a) is not satisfied, then the eigenvalues are eithercomplex or repeated. Do you think that I(t) 0 and V(t) 0 as t in these casesas well?Hint: In part (c), one approach is to change the system (i) into a single second orderequation. We also discuss complex and repeated eigenvalues in Sections 7.6 and 7.8.