Systems of first-order linear ODEs
General · Mathematics
Study notes
Q: Solve x' = 2x+y, y' = x+2y (system). A = [[2,1],[1,2]]. Eigen: 3 ((1,1)), 1 ((1,-1)). General: c₁e^(3t)(1,1)+c₂e^t(1,-1). So x = c₁e^(3t)+c₂e^t, y = c₁e^(3t)-c₂e^t! Check: x' = 3c₁e^(3t)+c₂e^t = 2x+y ✓! (Eigenvectors decouple!)