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IV. Differentialligninger – 2. 1. ordens ligninger 326 En partikulær løsning er en differentiabel funktion y(x) som opfylder y ′ (x) = a(x)y(x) + b(x) Den fuldstœndige løsning er en angivelse af alle løsninger. Ligningen dy dx = a(x)y kaldes homogen og er den homogene part af den inhomogene, b = 0, ligning ovenfor. 2.4. Superposition ☞ [LA] 14 Lineær differentialligning Sætning 22 Hvis z1(x),z2(x) er løsninger til den homogene lineœre differentialligning så er enhver linearkombination også en løsning. dy = a(x)y dx z(x) = C1z1(x) + C2z2(x) 2.5. Superposition ☞ [LA] 14 Lineær differentialligning Sætning 22 - fortsat Hvis z0(x) er en løsning til den inhomogene lineœre differentialligning så er enhver løsning af formen dy = a(x)y + b(x) dx y(x) = z(x) + z0(x)
IV. Differentialligninger – 2. 1. ordens ligninger 327 hvor z(x) er en løsning til den homogene part af systemet. 2.6. 1. ordens lineær ligning ☞ [LA] 14 Lineær differentialligning Sætning 23 Den lineœre ligning med konstante koefficienter har fuldstœndig løsning givet ved a = 0: a = 0: hvor C er arbitrœr. dy = ay + b dx y(x) = C + bx y(x) = Ce ax − b a 2.7. 1. ordens lineær ligning ☞ [LA] 14 Lineær differentialligning Sætning 24 Den homogene lineœre ligning dy = a(x)y dx har fuldstœndig løsning y(x) = Ce A(x)
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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