Theorem: If A has a repeated real root λ with corresponding eigenvector K₁ then all solutions of X' = A X are of the form: W x ν = (*) -K et K₁₂ (te K₁ + e₁t W2 where (A - I) (B)- = K₁. Theorem: Let λ₁ = 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: ()- That is: }) y = c₁ (B₁ αι cos ẞt - B2 sin ẞt) eat + 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 K₁ then all solutions of X' = A X are of the form: W x ν = (*) -K et K₁₂ (te K₁ + e₁t W2 where (A - I) (B)- = K₁. Theorem: Let λ₁ = 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: ()- That is: }) y = c₁ (B₁ αι cos ẞt - B2 sin ẞt) eat + 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.
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
Question
Using the attached theorms, Solve dx/dt =2x+3y , dy/dt=6x+5y with initial conditions x(0) = 8, y(0) = 1.
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