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course MTH-279

08/03 around 10:55 PM

Query 24 Differential Equations*********************************************

Question: Verify Abel's Theorem in the interval (-infinity, infinity) for

y ' = [ 6, 5; -7, -6] * y

whose solutions are

y_1 = [ 5 e^-t; -7 e^-t ]

y_2 = [ e^t; - e^t ]

with t_0 = -1

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y1 & y2 -> [5e^-t, e^t ; -7e^-t, -e^t]

W(t) = (5e^-t * -e^t) - (-7e^-t * e^t)

= (5e^-t * -e^t) + (7e^-t * e^t)

= (-5e^0) + (7e^0)

= -5 + 7

= 2

so, W(t) does not equal zero.

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Abel’s theorem states if y1, y2, … yn is a set of solutions of y’ = P(t)y, a

knowing Wronskian of solutions then W(t) satisfies the scalar differential equation.

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W’(t) = Pn-1(t)Wt

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W(t) = W(t0)e - int(t0 to t)Pn-1(s) ds

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W’(t) = tr[P(t)] * W(t)

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W’(t) = 0 * 2

W’(t) = 0

Confidence rating: 2

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Given Solution:

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