Query 24

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course MTH 279

The last question on the Query was blank when I opened it at:http://vhcc2.vhcc.edu/dsmith/GenInfo/qa_query_etc/differential_equations/query_24.htm

I left it as is, and so the last part that says ""question:"" and etc, was blank to begin with." "Query 24 Differential Equations

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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:

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

A = p(t) = [6,5; -7,-6]

W(t) = -5 + 7 = 2

tr[A] = 0

W’(t) = 0 = tr[A]*W(t) = 0*2 = 0

W(t_0) = W(-1) = 2

Since this relationship is true, and the Wronskian ≠ 0, then we can know that the Wronskian is never zero on (-∞,∞). This means that y_1 and y_2 form a fundamental set.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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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:

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

W(t) = det(y) = 3e^(3t).

The Wronskian is non-zero at all values of t, so the solutions form a fundamental set.

W’(t) = tr[A]W(t) = tr[A]*3e^(3t) = 9e^(3t)

tr[A] = 3

Ψ(t) = [y_1,y_2] = [5,2e^(3t); 1, e^(3t)]

Ψ’ = A*Ψ

A = Ψ’*Ψ^(-1)

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

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

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

tr[A] = -2+5 = 3

This is consistent.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question:

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Your solution:

confidence rating #$&*:

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

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:"

&#Good responses. Let me know if you have questions. &#