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IV. Differentialligninger – 3. Generelle metoder 346 3. Generelle metoder 3.1. Oversigt ☞ [S] 7.2, 7.5, 7.6; [LA] 17, 18 Nøgleord og begreber ✌ Eksistens og entydighed ✌ Retningsfelt ✌ Eulers metode ✌ Hastighedsfelt ✌ Stabilitet 3.2. Ligning og løsning ☞ [LA] 17 Generel ligning Definition 1 Lad I,J være åbne intervaller og F(x,y) : I × J → R en reel funktion. En løsning til differentialligningen dy = F(x,y) dx er en differentiabel funktion y(x) : I ′ → J på et åbent delinterval I ′ ⊆ I, som indsat giver y ′ (x) = F(x,y(x)), x ∈ I ′ 3.3. Eksistens og entydighed ☞ [LA] 17 Generel ligning
IV. Differentialligninger – 3. Generelle metoder 347 Sætning 32 Antag at F(x,y) er kontinuert og ∂F ∂y (x,y) eksisterer og er kontinuert i I ×J. For et givet (x0,y0) ∈ I × J findes entydigt bestemt et maximalt delinterval I ′ ⊆ I og en differentiabel funktion y(x) : I ′ → J, som er en løsning til differentialligningen og opfylder dy = F(x,y) dx y(x0) = y0 3.4. Eksistens og entydighed ☞ [LA] 17 Generel ligning Bemærkning 1 Den udvidede ligning dy dx = F(x,y),y(x0) = y0 kaldes et begyndelsesværdiproblem. Eksistens- og entydighedssætningen for begyndelsesværdiproblemer har en naturlig og vigtig udvidelse til differentialligningssystemer. 3.5. Eksistenseksempel ☞ [LA] 17 Generel ligning Eksempel 1
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Litteratur • Arnold, Ordinary dif
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