Query_24

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course Mth 279

5/17

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

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Your solution:

W(t) = det[psi(t)]

psi(t) = [ 5e^(-t), e^t; -7e^(-t), -e^t ]

W(t) = (5e^(-t))(-e^t) - (e^t)(-7e^t) = -5 + 7 = 2

W '(t) = should be zero.

Abel's Theorem:

W '(t) = tr[p(t)]*W(t)

tr[p(t)] = (6 + -6) = 0

W '(t) = tr[p(t)]*W(t)

W '(t) = 0*2 = 0

This turns out to be correct!

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:

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Question: y ' = A y, with solutions

y_1 = [5; 1]

y_2 = [2 e^(3 t), e^(3 t) ]

Verify that this constitutes a fundamental set.

Find Tr(A).

Show that

psi(t) = [y_1, y_2]

satisfies

psi ' = A * psi

Find A by finding psi ' * psi^-1

Is the result consistent with your result for the trace of A?

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Your solution:

psi(t) = [ 5, 2e^(3t); 1, e^(3t)]

W(t) = 5e^(3t) - 2e^(3t) = 3e^(3t)

since W(t) is not zero, its a fundamental set.

W ' (t) = tr[A(t)]*W(t)

W ' (t) = 9e^(3t)

9e^(3t) = tr[A(t)]*(3e^(3t))

tr[A(t)] = 3

Finding A = psi ' (t) * psi^(-1)(t)

psi ' (t) = [ 0, 6e^(3t); 0, 3e^(3t)]

and

psi^(-1)(t) = (1/3)e^(-3t) * [ e^(3t), -2e^(3t); -1, 5 ]

=[ (1/3), (-2/3); -1/(3e^(3t)), 5/(3e^(3t)) ]

A(t) = [ 0, 6e^(3t); 0, 3e^(3t)]*[ (1/3), (-2/3); -1/(3e^(3t)), 5/(3e^(3t)) ] = [ -2, 10; -1, 5 ]

tr[A(t)] = -2+5 = 3

psi ' (t) = A(t)psi(t)

[ -2, 10; -1, 5 ]*[ 5, 2e^(3t); 1, e^(3t)] = [ 0, 6e^(3t); 0, 3e^(3t)]

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:

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Question:

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Your solution:

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confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:

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Self-critique (if necessary):

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Self-critique rating:

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Self-critique (if necessary):

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Self-critique rating:

#*&!

&#Very good responses. Let me know if you have questions. &#