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.