Query 24

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course MTH 279

5:09 pm 5/3

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: 

P = [ 6, 5; -7, -6] --> Tr[P] = 0

W(t)= W(t_0) * 1

W = det [ 5 e^-t, e^t; -7 e^-t , -e^t]

W = 2 & W(t_0) = det [5e^t, e^-t; -7e^t, -e^-t] = 2

 

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Given Solution: 

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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: 

W = det [5, 2e^3t; 1, e^3t] = 3e^3t

W' = 9e^3t

W' = Tr[A] W(t) --Tr[A] = 3

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

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

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

A = psi' * psi^-1 = [0, 6e^3t; 0, 3e^3t] * [1/3, -2; -1/(3e^3t), 5/(3e^3t)] = [-2, 10; -1, 5]

Tr[A] = -2 + 5= 3 so these are consistent

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Given Solution: 

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Self-critique (if necessary):

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Self-critique rating:"

Self-critique (if necessary):

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