6.Consider the electric circuit illustrated below. Let I be the current through the inductorand V be the voltage across the capacitor. Then I and V satisfy the linear system:LIPdtR1I +L().VR1R2APdt1I +VCR2Let R11N, 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).

“Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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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