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lecture_15 (dragged) 2 - MA 36600 LECTURE NOTES FRIDAY...

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MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 20 3 We explain how this is related to Abel’s Theorem. Using the Quadratic Formula, we see that the roots of the characteristic equation are r 1 = b 2 a b 2 4 a c 2 a r 2 = b 2 a + b 2 4 a c 2 a = r 1 + r 2 = b a r 2 r 1 = b 2 4 a c a Hence the Wronskian is in the form W ( t ) = C exp b a t in terms of C = b 2 4 a c a . Since b 2 4 a c 0 by assumption, we see that this function is nonzero if and only if b 2 4 a c > 0. Applications of Abel’s Theorem. First, we show that if W ( t 0 ) = 0 for some t 0 then y = c 1 y 1 + c 2 y 2 is the general solution to the di ff erential equation. To see why, we can consider some initial conditions y ( t 0 ) = y 0 and y ( t 0 ) = y 0 . Since W 0 = W ( t 0 ) = 0 we can solve for c 1 and c 2 . Note that W ( t 0 ) = 0 precisely when the constant C = 0. Second, we give a method for finding solutions to the di ff erential equation. Say that we know one solution y 1 = y 1 ( t ) is one solution. Define the function
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