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.

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