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# W x 0 y1 y2 xi i w x 0 w x 0 xi i y

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Unformatted text preview: f c∈R g f y = c1 y1 + c2 y2 y = c1 ky2 + c2 y2 = (kc1 + c2 )y2 I −→ R f (x) = c · g (x) g y1 y1 f g x∈I y1 g (x) = c · f (x) ✷ y ￿￿ +p(x)y ￿ +q (x)y = 0 y2 y2 y ￿￿ + p(x)y ￿ + q (x)y = 0 I⊂R W (x) ￿ ￿ ￿y1 (x) y2 (x)￿ ￿ ￿ ￿ W (x) := ￿ ￿ ￿ ￿y1 (x) y2 (x)￿ = y1 (x) · y2 (x) − y2 (x) · y1 (x). W (x) = 0 y1 y2 x∈I I W (x) ￿= 0 W (x) ￿= 0 x∈I I ✷ y ￿￿ + p(x)y ￿ + q (x)y = 0 x0 ∈ I W (x0 ) y1 (x) = sin(x) y2 (x) = cos(x) y ￿￿ + y = 0 R ￿ ￿ ￿y1 (x) y2 (x)￿ ￿ ￿ 2 2 ￿ W (x) = ￿ ￿ ￿ ￿y1 (x) y2 (x)￿ = y1 (x) · y2 (x) − y2 (x) · y1 (x) = − sin x − cos x = −1 ￿= 0 x∈R x0 ∈ R 0 ✷ ￿= 0 1 16 y1 (x) = 1 − x3 + x + ... 6 180 y2 (x) = x − 14 17 x+ x + ... 12 504 y ￿￿ + xy = 0 W (0) ￿ ￿ W (0) = y1 (0) · y2 (0) − y2 (0) · y1 (0) = 1 · 1 − 0 · 0 = 1 ￿= 0 ✷ I⊂R y1 y ￿￿ + p(x)y ￿ + q (x)y = 0 y2 y1 y2 y (x) = c1 · y1 (x) + c2 · y2 (x), c1 c2 y ￿￿ + p(x)y ￿ + q (x)y = f (x), p(x) q (x) I⊂R f (x) y1 (x) (∗) y2 (x) yp (x) y ￿￿ + p(x)y ￿ + q (x)y = 0. (∗) y (x) = c1 · y1 (x) + c2 · y2 (x) + yp (x), c1 ϕ(x) c2 (∗) ￿ ￿￿...
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