Linear System of Differential Equations

dxdt

=ax+by dydt

=cx+dy

dYdt

=AY → ddt [ xy ]=[a b

c d ] [ xy ]

CASE 1: Two Distinct Real Eigenvalues (λ1, λ2 with Eigenvectors V1,V2)

General Solution: k 1V 1 ek1 t+k2V 2e

k2 t

CASE 2: Complex Conjugate Eigenvalues ( λ1,2 = α±βi )

General Solution: k 1eαt [ xℜ

yℜ]+k2eαt [ xℑ

yℑ]CASE 3: Repeated Real Eigenvalues (arbitrary eigenvalue V0=(x0,y0))

General Solution: k 1eλ1 tV 0+k2t e

λ1tV 1

Where V1=(A-Iλ)V0