midterm_2_review (dragged) 8

midterm_2_review (dragged) 8 - 8 MA 36600 MIDTERM#2 REVIEW...

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Unformatted text preview: 8 MA 36600 MIDTERM #2 REVIEW §4.4: The Method of Variation of Parameters. • Consider the nth order nonhomogeneous linear differential equation dn Y dn−1 Y dY + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) Y = G(t). n dt dt dt Say that {y1 , y2 , . . . , yn } is a fundamental set of solutions to the homogeneous equation. Then a solution to the nonhomogeneous equation is ￿t G(τ ) Y (t) = u1 (t) y1 (t) + u2 (t) y2 (t) + · · · + un (t) yn (t) = K (τ , t) dτ P0 (τ ) P0 (t) where the ui (t) satisfy the system of first order differential equations du1 du2 dun y1 (t) + y2 (t) + ··· + yn (t) dt dt dt du1 du2 dun (1) (1) (1) y1 (t) + y2 (t) + ··· + yn (t) dt dt dt du1 dt du1 (n−1) y1 (t) dt (n−2) y1 (t) du2 dt du2 (n−1) + y2 (t) dt (n−2) + y2 (t) + ··· + ··· dun dt dun (n−1) + yn (t) dt (n−2) + yn (t) = 0 = 0 . . . = 0 = G(t) P0 (t) and K (τ , t) is the ratio of determinants ￿ ￿￿ ￿ y1 (τ ) y2 (τ ) y2 (τ ) ··· yn (τ ) ￿ ￿ y1 (τ ) ￿ ￿￿ ￿ (1) ￿ ￿ (1) (1) (1) (1) ￿ y (τ ) y2 (τ ) y2 (τ ) ··· yn (τ ) ￿￿ ￿ y1 (τ ) ￿1 ￿￿ ￿ ￿￿ . . ￿ ￿￿ . . . .. . . . . . K (τ , t) = ￿ ￿￿ . . . . . . ￿ ￿￿ ￿ (n−2) ￿ ￿ (n−2) (n−2) (n−2) (n−2) ￿y (τ ) y2 (τ ) (τ ) y2 (τ ) · · · yn (τ )￿ ￿y1 ￿1 ￿￿ ￿ ￿ ￿ (n−1) (n−1) ￿ y1 (t) y2 (t) ··· yn (t) ￿ ￿y1 (τ ) y2 (τ ) ￿ ￿￿ ￿ ··· ··· .. . ··· ··· ￿ W y1 , y2 , ..., yn (τ ) ￿ yn (τ ) ￿ ￿ ￿ (1) yn (τ ) ￿ ￿ ￿ . ￿ . ￿. . ￿ ￿ (n−2) yn (τ )￿ ￿ ￿ (n−1) yn (τ )￿ ￿ ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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