Assignment 26

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course Mth 279

8/5

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:

1) find W(t)?

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

2) find tr[P]?

tr[P]= 6+(-6) = 0

3) Verify Abel’s Theorem

use

W(t) = W(t_0)e^(int(tr[P]))

=2e^(int(0))

=2e^0

W(t)=2

which is exactly the same as what we got in step 1 so Abel’s Theorem verified

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:

1) W(t) = [5, 2e^(3t); 1, e^(3t)] = 5e^(3t) - 2e^(3t) = 3e^(3t), yes constitutes fund set

2) Tr[A] How do you find Tr[A} with only y_1 and y_2????

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W ' (t) = tr(A) * W(t).

You have W and you can easily get W ', then solve the matrix equation for A. You are guaranteed a solution because the determinant of W is nonzero, hence W is invertible.
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3) psi(t) = [5, 2e^(3t); 1, e^(3t)]

psi’ = [0, 6e^(3t); 0, 3e^(3t)]

4) psi^-1 = 1/(ad-bc) [d, -b; -c, a] = 1/(5e^(3t)-2e^(3t))[e^(3t), -2e^(3t); -1, 5]

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

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

A = [-2, 10; -1,5]

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

Now sure what I am comparing too to see if it is consistent or not???

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Had you found the trace of A previously, you would be able to compare that result to this one.
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