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Consider the electric circuit illustrated below. Let I be the current through the inductor and V be the voltage across the capacitor. Then I and V satisfy the linear system: L IP dt R1 I+ (-)» RE R2 AP dt I + V CR2, Let R1 = 1N, R2=N, L = 2 H, and C = F. Answer the following questions about the system: (a) Express the system in matrix notation: ' = PE (b) Verify that a1 is an eigenvector of P. Identify the corresponding eigenvalue A1. %3D

Consider the electric circuit illustrated below. Let I be the current through the inductor and V be the voltage across the capacitor. Then I and V satisfy the linear system: L IP dt R1 I+ (-)» RE R2 AP dt I + V CR2, Let R1 = 1N, R2=N, L = 2 H, and C = F. Answer the following questions about the system: (a) Express the system in matrix notation: ' = PE (b) Verify that a1 is an eigenvector of P. Identify the corresponding eigenvalue A1. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. Consider the electric circuit illustrated below. Let I be the current through the inductor and V be the voltage across the capacitor. Then I and V satisfy the linear system: L IP dt R1 I + L (). V R1 R2 AP dt 1 I + V CR2 Let R1 1N, R2 = N, L = 2 H, and C = ? F. Answer the following questions about the system: (a) Express the system in matrix notation: T' = Pa (6) (b) Verify that ī = is an eigenvector of P. Identify the corresponding eigenvalue A1. (c) Given that A2 = -2 is an eigenvalue of P, identify an eigenvector 2 that corresponds to A2. (d) Find the general solution to the system of differential equations. (e) Show that I(t) → 0 and V (t) → 0 as t → ∞, regardless of the initial values of I(0) and V(0).
Transcribed Image Text:6. Consider the electric circuit illustrated below. Let I be the current through the inductor and V be the voltage across the capacitor. Then I and V satisfy the linear system: L IP dt R1 I + L (). V R1 R2 AP dt 1 I + V CR2 Let R1 1N, R2 = N, L = 2 H, and C = ? F. Answer the following questions about the system: (a) Express the system in matrix notation: T' = Pa (6) (b) Verify that ī = is an eigenvector of P. Identify the corresponding eigenvalue A1. (c) Given that A2 = -2 is an eigenvalue of P, identify an eigenvector 2 that corresponds to A2. (d) Find the general solution to the system of differential equations. (e) Show that I(t) → 0 and V (t) → 0 as t → ∞, regardless of the initial values of I(0) and V(0).
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