De Linear Systems Cheat Sheet
1
Linear System of Differential Equations dx dt =ax +by dy dt =cx +dy dY dt =AY → d dt [ x y ] = [ a b c d ][ x y ] CASE 1: Two Distinct Real Eigenvalues (λ 1, λ 2 with Eigenvectors V 1 ,V 2 ) General Solution: k 1 V 1 e k 1 t + k 2 V 2 e k 2 t CASE 2: Complex Conjugate Eigenvalues ( λ 1,2 = α±βi ) General Solution: k 1 e αt [ x ℜ y ℜ ] +k 2 e αt [ x ℑ y ℑ ] CASE 3: Repeated Real Eigenvalues (arbitrary eigenvalue V 0 =(x 0 ,y 0 )) General Solution: k 1 e λ 1 t V 0 + k 2 te λ 1 t V 1 Where V 1 =(A-Iλ)V 0
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a short list of Differential equations notes, its incomplete
Transcript of De Linear Systems Cheat Sheet
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
