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IV. Differentialligninger – 2. 1. ordens ligninger 336 Fra Særning 1.3 fås den fuldstændige løsning på vektorform giver dette y(x) = eller udtrykt ved egenvektorerne y1(x) = C1e λ1x , y2(x) = C2e λ2x y1(x) = y2(x) C1eλ1x C2e λ2x = C1 e λ1x y(x) = C1e λ1x e1 + C2e λ2x e2 0 + C2 0 eλ2x 2.24. Lineært system ☞ [LA] 15 Lineært system Sætning 27 Betragt 2 × 2-matricen A = (aij) og 2-søjlen y(x) = (yi(x)) samt det homogene lineœre differentialligningssystem dy = Ay dx Hvis u er en egenvektor for A med egenvœrdi λ, så er løsninger, hvor C er arbitrær. y(x) = Ce λx u 2.25. Lineært system ☞ [LA] 15 Lineært system
IV. Differentialligninger – 2. 1. ordens ligninger 337 Sætning 28 Betragt 2 × 2-matricen A = (aij) og 2-søjlerne b = (bi), y(x) = (yi(x)) samt det lineœre differentialligningssystem dy = Ay + b dx En konstant funktion y(x) = v er en løsning, hvis Av = −b. Hvis yderligere u er en egenvektor for A med egenvœrdi λ, så er y(x) = Ce λx u + v løsninger, hvor C er arbitrær. 2.26. Lineært system ☞ [LA] 15 Lineært system Sætning 29 Betragt 2 × 2-matricen A = (aij) og 2-søjlen y(x) = (yi(x)) samt det homogene lineœre differentialligningssystem dy = Ay dx Hvis y0 = C1u1 + C2u2 er en linearkombination af egenvektorer for A, med egenvœrdier λ1,λ2, Auj = λjuj, så er en løsning, der opfylder y(0) = y0. y(x) = C1e λ1x u1 + C2e λ2x u2
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Forord "Slides" til forelæsningern
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