X' = AX has a saddle. ALWAYS where D = det A and I ALWAYS SOMETIMES ace(A), and A is a 2 x 2 matrix, then the system (e) If x(t) and y(t) are solutions to the equation x" + px' + qx = sint and x'(0) = y'(0), then x(t)= y(t), for all t. SOMETIMES NEVER NEVER
X' = AX has a saddle. ALWAYS where D = det A and I ALWAYS SOMETIMES ace(A), and A is a 2 x 2 matrix, then the system (e) If x(t) and y(t) are solutions to the equation x" + px' + qx = sint and x'(0) = y'(0), then x(t)= y(t), for all t. SOMETIMES NEVER NEVER
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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Transcribed Image Text:(d) If D = T²+1, where D = det A and T = trace(A), and A is a 2 × 2 matrix, then the system
X' = = AX has a saddle.
ALWAYS
SOMETIMES
ALWAYS
(e) If x(t) and y(t) are solutions to the equation x" + px' + qx = sint and x'(0) = y'(0), then
x(t) = y(t), for all t.
NEVER
SOMETIMES
NEVER
![1.
For each of the following statements, determine if the conclusion ALWAYS follows
from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the
conclusion is NEVER true given the assumptions.
(a) If X(t) is a non-zero solution to the linear system X' = AX, where A is a 2 × 2 matrix with
det A = 0, and X(1) = X(0), then A has imaginary eigenvalues.
ALWAYS
ALWAYS
(b) If x(t) is a solution to a damped unforced mass-spring oscillator equation, then x(t) → 0,
as t→→∞.
(c) If X(t) =
ALWAYS
SOMETIMES
[0]
SOMETIMES
NEVER
SOMETIMES
NEVER
is a solution to the system X' = AX, then x(t)² + y(t)² → ∞, as t → 1.
NEVER](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c9b130f-7b54-4cdb-81ba-a147d154821a%2F63c3b7af-152b-4e0b-a949-7fb36a5dfe6d%2F4cow6fm_processed.png&w=3840&q=75)
Transcribed Image Text:1.
For each of the following statements, determine if the conclusion ALWAYS follows
from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the
conclusion is NEVER true given the assumptions.
(a) If X(t) is a non-zero solution to the linear system X' = AX, where A is a 2 × 2 matrix with
det A = 0, and X(1) = X(0), then A has imaginary eigenvalues.
ALWAYS
ALWAYS
(b) If x(t) is a solution to a damped unforced mass-spring oscillator equation, then x(t) → 0,
as t→→∞.
(c) If X(t) =
ALWAYS
SOMETIMES
[0]
SOMETIMES
NEVER
SOMETIMES
NEVER
is a solution to the system X' = AX, then x(t)² + y(t)² → ∞, as t → 1.
NEVER
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