Question

-2 1 (1 point) Find the most general real-valued solution to the linear system of differential equations z' xi(t) - = C1 + C2 22(t)

Answer 1

Given that the linear system of differential equations a' -2 -1 -4 → X'= Añ, where A21-2 1 L-1-4 The characteristie equation of A is det (A-A 1) -0. 1-2-2 1 = 0 -1 – 4-2 > 2-1) 6-4-) +120 +) (4+2)+1=0 8+62 +ar+1=0 2Y+62 +920 → @ +3)=0. :.2=-3 is an eigenvalue of A with multiplicity Let's find the eigen vector of A corres- ponding to this eigenvalue a= be an eigenvector of A corresponding to the eigenvalue X=-3. 2. -3 :: V = 21 122 1 .'.-2+3 -1 - 4+3 → +22=0 and -no-nero an+h2o. :.=-22 ut n2 = S, SER » No-s. vi n2 s

::M -1] We now find the eigen vector v2 = Cell in such an way that [:-]®]-[] → P1+12=-1 and - P1-P2=1 » Pltl22-1. >p =-1-P2 Let P2=0. :: P = -1 .. vz- ()

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