Theorem: If A has a repeated real root λ with corresponding eigenvector K, then all solutions of X' = A X are of the form: W1 ( ; ) - = C1 et K₁₂ (te K₁+ e^t λι W 2 W1 where (A - I) = K₁. W 2 Theorem: Let₁ = α + i ẞ be a complex eigenvalue of the coefficient matrix A with corresponding eigenvector K₁ = B₁ + i B2. Then all solutions of X' = A X are of the form: (ţ♬ ) - χ That is: y αι е |= c₁ ( B₁ cos ßt - B₂ sin ẞt) eª² + c₂ ( B₂ cos ẞt + B₁ sin ẞt) eat. |= c₁ (Re(K₁) cos ẞt - Im(K₁) sin ẞt) eat + c2 ( Im(K₁) cos ẞt + Re(K₁) sin ẞt) eat.
Theorem: If A has a repeated real root λ with corresponding eigenvector K, then all solutions of X' = A X are of the form: W1 ( ; ) - = C1 et K₁₂ (te K₁+ e^t λι W 2 W1 where (A - I) = K₁. W 2 Theorem: Let₁ = α + i ẞ be a complex eigenvalue of the coefficient matrix A with corresponding eigenvector K₁ = B₁ + i B2. Then all solutions of X' = A X are of the form: (ţ♬ ) - χ That is: y αι е |= c₁ ( B₁ cos ßt - B₂ sin ẞt) eª² + c₂ ( B₂ cos ẞt + B₁ sin ẞt) eat. |= c₁ (Re(K₁) cos ẞt - Im(K₁) sin ẞt) eat + c2 ( Im(K₁) cos ẞt + Re(K₁) sin ẞt) eat.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Using the attached theorms, Solve the following parts:
a) dx/dt =7x-y , dy/dt=5x+3y
b)dx/dt =-x+3y , dy/dt=-3x+5y
Transcribed Image Text:Theorem: If A has a repeated real root λ with corresponding eigenvector K, then all
solutions of X' = A X are of the form:
W1
( ; ) -
=
C1
et K₁₂ (te K₁+ e^t
λι
W 2
W1
where (A - I)
= K₁.
W 2
Theorem: Let₁ = α + i ẞ be a complex eigenvalue of the coefficient matrix A with corresponding
eigenvector K₁ = B₁ + i B2. Then all solutions of X' = A X are of the form:
(ţ♬ ) -
χ
That is:
y
αι
е
|= c₁ ( B₁ cos ßt - B₂ sin ẞt) eª² + c₂ ( B₂ cos ẞt + B₁ sin ẞt) eat.
|= c₁ (Re(K₁) cos ẞt - Im(K₁) sin ẞt) eat + c2 ( Im(K₁) cos ẞt + Re(K₁) sin ẞt) eat.
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