Consider the preceding system of differential equations (37). a. Find a condition on R1

Chapter 7, Problem 25

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Consider the preceding system of differential equations (37). a. Find a condition on R1, R2, C, and L that must be satisfied if the eigenvalues of the coefficient 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 both 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 either complex or repeated. Do you think that I (t) 0 and V(t) 0 as t in these cases as well? Hint: In part c, one approach is to change the system (37) into a single second-order equation. We also discuss complex and repeated eigenvalues in Sections 7.6 and 7.8.

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