MATH 320 - Ordinary Differential Equations
Section 3.5: Repeated and Zero Eigenvalues
Friday, April 16, 2004
Example 1 (Repeated Eigenvalues)
dY
dt =Ã2 1
−1 4 !Y
Linear System with A such that
a=2,b=1,c=–1, d=4
–4
–2
2
4
y
–4 –2 2 4
x
Theorem 1 (General Solution:)
Suppose dY
dt =AYis a linear system in which the 2×2matrix Ahas a repeated real eigenvalue λ
but only one line of eigenvectors. Then the general solution has the form
Y(t) = eλtV0+teλtV1
where V0= [x0, y0]T, the initial value vector, and V1is determined from V0by
V1= (A−λI)V0
NOTE: If V0is an eigenvector of A, then V1= 0, and if V0is not an eigenvector of A, the V1is!
Prove this “NOTE”:
