Systems of Frst-order differential equations163orx(t)=c1erty(t)=c2ertExample 4.10Let˙x=x−y˙y=x+3yThen±˙x˙y²=±1−113²±xy²withA=±1−1²,det(A)=4,A−λI=±1−λ−1−λ²Hence, det(A−λI)=λ2−4λ+4=(λ−2)2, with rootλ=r=2 (twice).Usingλ=r=2A−rI=±−1−111²and(A−rI)±xy²=±−1−1xy²which implies−x−y=0. Given we normalisextox=1,theny=−1. The frstsolution is thene2t±1−1²To obtain the second solution we might think oF proceeding as in the singlevariable case, but this is not valid (see Boyce and DiPrima 1997, pp. 390–6). Whatwe need to use is
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