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 + ()► V R1 R2 (). dV 1 I + V CR2 dt C Let R1 = 10, R2 -n, L – 2 H, and C = - F. Answer the following questions about the system: (a) Express the system in matrix notation: T = PF (). (b) Verify that = is an eigenvector of P. Identify the corresponding eigenvalue A1. (c) Given that12 = -2 is an eigenvalue of P, identify an eigenvector 2 that corresponds to X2. (d) Find the general solution to the system of differential equations. (e) Show that I(t) + 0 and V(t) → 0 as t → 0, regardless of the initial values of I(0) and V(0).
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 + ()► V R1 R2 (). dV 1 I + V CR2 dt C Let R1 = 10, R2 -n, L – 2 H, and C = - F. Answer the following questions about the system: (a) Express the system in matrix notation: T = PF (). (b) Verify that = is an eigenvector of P. Identify the corresponding eigenvalue A1. (c) Given that12 = -2 is an eigenvalue of P, identify an eigenvector 2 that corresponds to X2. (d) Find the general solution to the system of differential equations. (e) Show that I(t) + 0 and V(t) → 0 as t → 0, regardless of the initial values of I(0) and V(0).
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter11: Systems Of Equations
Section11.5: Cramer's Rule
Problem 35PS
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Question
Need parts c, d and e please
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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