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I. Differentiation – 2. Partielle afledede 50 Ligningen kaldes Laplaces ligning. Ligningen kaldes bølgeligningen. ∂ 2 u ∂x2 + ∂2u ∂y 2 = 0 ∂2u ∂t2 = a2 ∂2u ∂x2 2.29. Laplaces ligning ☞ [S] 11.3 Partial derivatives Eksempel 8 Funktionen u(x,y) = e x sin y er løsning til Laplaces ligning Løsning giver uxx + uyy = 0 ux = e x siny, uxx = e x siny uy = e x cos y, uyy = −e x siny uxx + uyy = 0 2.30. Bølgeligningen ☞ [S] 11.3 Partial derivatives
I. Differentiation – 2. Partielle afledede 51 Eksempel 9 Funktionen u(t,x) = sin(x − at) er løsning til bølgeligningen Løsning giver utt = a 2 uxx ut = −acos(x − at), utt = −a 2 sin(x − at) ux = cos(x − at), uxx = −sin(x − at) utt = a 2 uxx 2.31. Test Laplaces ligning ☞ [S] 11.3 Partial derivatives Test Funktionen f(x,y) = 3x + 5y + 10 er en løsning til Laplace’s ligning Løsning Udregningen giver ∂ 2 f/∂x 2 + ∂ 2 f/∂y 2 = 0. fx = 3, fxx = 0, fy = 5, fyy = 0 fxx + fyy = 0 Afkryds: ja nej
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1. Grafiske/numeriske metoder IV Di
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V. Matricer - 4. Determinanter = (
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Litteratur • Arnold, Ordinary dif
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0-række/søjle, 423 n-te afsnitssu
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