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IV. Differentialligninger – 2. 1. ordens ligninger 338 2.27. Lineært system ☞ [LA] 15 Lineært system Sætning 30 Betragt 2 × 2-matricen A = (aij) og 2-søjlen y(x) = (yi(x)) samt det homogene lineœre differentialligningssystem dy = Ay dx Hvis matricen U med søjler u1,u2 diagonaliserer A med egenvœrdier λ1,λ2, Auj = λjuj, så er den fuldstændige løsning givet ved hvor C1,C2 er arbitrœre. y(x) = C1e λ1x u1 + C2e λ2x u2 2.28. Lineært system ☞ [LA] 15 Lineært system Sætning 31 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 matricen U med søjler u1,u2 diagonaliserer A med egenvœrdier λ1,λ2, Auj = λjuj, så er den fuldstændige løsning givet ved hvor C1,C2 er arbitrœre. y(x) = C1e λ1x u1 + C2e λ2x u2 + v
IV. Differentialligninger – 2. 1. ordens ligninger 339 2.29. Opgave ☞ [LA] 15 Lineært system Opgave 1 Betragt differentialligningssystemet y ′ 1 = y1 + y2 y ′ 2 = 8y1 − y2 Det oplyses, at vektoren u = (1,2) er en egenvektor for matricen 1 1 A = 8 −1 Angiv den løsning y(x) = (y1(x),y2(x)) der opfylder y(0) = u, altså (y1(0),y2(0)) = (1,2). 2.30. Opgave ☞ [LA] 15 Lineært system Opgave 1 - fortsat Egenværdien λ = 3 fås af udregningen I følge [LA] Sætning 27 er løsninger for alle valg af C. Au = 1 1 1 8 −1 2 y(x) = Ce 3x = 1 2 3 = 3u 6 2.31. Opgave ☞ [LA] 15 Lineært system
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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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