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lecture_23 (dragged) 1 - 2 MA 36600 LECTURE NOTES:...

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Unformatted text preview: 2 MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 11 This is a linear differential equation if we can express ￿ ￿ n−1 ￿ G t, y, y (1) , . . . , y (n−1) = g (t) − p1 (t) y (n−1) − · · · − pn−1 (t) y ￿ − pn (t) y = g (t) − That is, if the differential equation is in the form j =0 pn−j (t) y (j ) . y (n) + p1 (t) y (n−1) + · · · + pn−1 (t) y ￿ + pn (t) = g (t). We remark equations in the following form are also linear: dn y dn−1 y dy P0 (t) n + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = G(t) dt dt dt =⇒ n ￿ j =0 Pn−j (t) y (j ) = G(t). Our goal is to show that if we can find (n + 1) functions y1 = y1 (t), . . . , yn (t) and Y = Y (t) such that (1) Each yi = yi (t) is a solution to the homogeneous equation n ￿ j =0 (j ) Pn−j (t) yi =0 for i = 1, 2, . . . , n. (2) The n × n determinant is nonzero: ￿ ￿ y1 (t) y2 (t) ··· ￿ ￿ (1) (1) ￿ y (t) y2 (t) ··· ￿1 ￿ ￿ W y1 , y2 , . . . , yn (t) = ￿ ￿ . . .. . . ￿ . . . ￿ ￿ (n−1) (n−1) ￿y (t) y2 (t) · · · 1 (3) Y = Y (t) is a solution to the nonhomogeneous equation n ￿ j =0 ￿ yn (t) ￿ ￿ ￿ (1) yn (t) ￿ ￿ ￿ ￿ (j −1) ￿ ￿ ￿ = ￿y ￿ ￿= 0. ￿ i . . ￿ . ￿ ￿ (n−1) yn (t)￿ Pn−j (t) Y (j ) = G(t). Then the general solution to the differential equation is ￿n ￿ ￿ y (t) = ci yi (t) + Y (t). i=1 Linear Operators. Given the linear differential equation P0 (t) dn y dn−1 y dy + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = G(t) n dt dt dt =⇒ Define the operator L[y ] = n ￿ i=1 Pn−j (t) y (j ) =⇒ n ￿ i=1 Pn−j (t) y (j ) = G(t). L[y ] = G(t). We will show that this is a linear operator. Given functions f = f (t) and g = g (t) as well as constants c1 and c2 , we have ￿ ￿￿ ￿ ￿ n n ￿ ￿￿ dj dj f dj g L c1 f + c2 g = Pn−j (t) j c1 f (t) + c2 g (t) = Pn−j (t) c1 j + c2 j dt dt dt i=1 i=1 ￿n ￿ ￿n ￿ ￿ ￿ dj f dj g = c1 Pn−j (t) j + c2 Pn−j (t) j dt dt i=1 i=1 = c1 L[f ] + c2 L[g ]. In particular, if {y1 , y2 , . . . , ym } is a set of functions satisfying L[yi ] = 0, then given the linear combination y (t) = c1 y1 (t) + c2 y2 (t) + · · · + cm ym (t) ...
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