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Theorem: If A has a repeated real root λ with corresponding eigenvector K₁ then all solutions of X' = A X are of the form: W1 x y ( 3 ) - G K + et K₁₂ (te K₁ + e₁t W2 W1 where (A - I) = K₁ . W2 Theorem: Let ₁ = 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: () - x That is: y αι |= c₁ ( B₁ cos ẞt - B₂ sin ẞt) eat + 2 (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 K₁ then all solutions of X' = A X are of the form: W1 x y ( 3 ) - G K + et K₁₂ (te K₁ + e₁t W2 W1 where (A - I) = K₁ . W2 Theorem: Let ₁ = 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: () - x That is: y αι |= c₁ ( B₁ cos ẞt - B₂ sin ẞt) eat + 2 (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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Using the attached theorms, Solve dx/dt =10x-20y , dy/dt=8x-18y 

Theorem: If A has a repeated real root λ with corresponding eigenvector K₁ then all solutions of X' = A X are of the form: W1 x y ( 3 ) - G K + et K₁₂ (te K₁ + e₁t W2 W1 where (A - I) = K₁ . W2 Theorem: Let ₁ = 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: () - x That is: y αι |= c₁ ( B₁ cos ẞt - B₂ sin ẞt) eat + 2 (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.
Transcribed Image Text:Theorem: If A has a repeated real root λ with corresponding eigenvector K₁ then all solutions of X' = A X are of the form: W1 x y ( 3 ) - G K + et K₁₂ (te K₁ + e₁t W2 W1 where (A - I) = K₁ . W2 Theorem: Let ₁ = 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: () - x That is: y αι |= c₁ ( B₁ cos ẞt - B₂ sin ẞt) eat + 2 (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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