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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 4 Variation of Parameters General Method. We explain how to find the general solution of the nonhomogeneous equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = f (t). Say that {y1 , y2 } is a fundamental set of solutions to the homogeneous equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = 0. Consider the function y (t) = u1 (t) y1 (t) + u2 (t) y2 (t). We have the derivatives y = u 1 y 1 + u 2 y2 ￿ ￿￿ ￿ ￿ ￿ y ￿ = u 1 y1 + u 2 y2 + u ￿ y1 + u ￿ y2 1 2 ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿￿ y = u 1 y1 + u 2 y 2 + u 1 y1 + u 2 y 2 + u ￿ y 1 + u ￿ y 2 1 2 This gives the expression f (t) = a(t) y ￿￿ + b(t) y ￿ + c(t) y ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ = u1 a(t) y1 + b(t) y1 + c(t) y1 + u2 a(t) y2 + b(t) y2 + c(t) y2 ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ + a(t) u￿ y1 + u￿ y2 + a(t) u￿ y1 + u￿ y2 + b(t) u￿ y1 + u￿ y2 . 1 2 1 2 1 2 Since y1 and y2 are solutions of a homogeneous equation, the middle row on the right-hand side is zero. We make the assumptions u￿ (t) y1 (t) + u￿ (t) y2 (t) = 0 1 2 f (t) a(t) We solve for u1 = u1 (t) and u2 = u2 (t) by considering a couple of first order differential equations. Multiply ￿ ￿ the first equation by y2 (t) (by −y1 (t)) and the second by −y2 (t) (by y1 (t), respectively) to find the following systems of equations: ￿ ￿ u￿ (t) y1 (t) + u￿ (t) y2 (t) = 1 2 ￿ y1 y2 u￿ (t) + 1 ￿ y2 y2 u￿ (t) = 2 0 ￿ ￿ −y1 y2 u￿ (t) + −y2 y2 u￿ (t) = − 1 2 W u￿ (t) 1 =− ￿ ￿ −y1 y1 u￿ (t) + −y1 y2 u￿ (t) = 1 2 f (t) y2 a(t) 0 ￿ y1 y2 u￿ (t) = 2 f (t) y1 a(t) W u￿ (t) = 2 ￿ y1 y1 u￿ (t) + 1 f (t) y1 a(t) f (t) y2 a(t) ￿ ￿ where we have denoted the function W = y1 y2 − y1 y2 . This is the Wronskian of y1 and y2 ; since {y1 , y2 } is a fundamental set of solutions the Wronskian is nonzero. Hence we find the first order differential equations u￿ (t) = − 1 f (t) y2 (t) a(t) W (t) and u￿ (t) = 2 f (t) y1 (t) . a(t) W (t) To recap, say that we wish to find the general solution to the differential equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = f (t). Perform the following steps: 1 ...
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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 University-West Lafayette.

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