Query 24

#$&*

course MTH 279

9 4/8

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:

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

W = -5e^-te^t + 7e^te^-t

W(-1) = 2

@&

Abel's Theorem says that W ' (t) is equal to the trace of the matrix p(t) = [6, 5; -7, -6], multiplied by W(t).

tr(p(t)) = 6 - 6 = 0

W(t) = W(0) = -5 + 7 = 2, so

W' (t) = 0.

Thus W ' (t) is equal to tr(p(t)) * W(t).

*@

confidence rating #$&*:

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

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

------------------------------------------------

Self-critique rating:

*********************************************

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?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

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

W = 5e^3t - 2e^3t = 3e^-t , which isn't 0 for any t, so this constitutes a fundamental set.

W' = tr(A) * W

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

tr(A) = 3

psi' = [0 , 6e^3t ; 0 , 3e^3t] = [a , b ; c , d] [5 , 2e^3t ; 1 , e^3t]

A = psi' * psi^-1 = [0 , 6e^3t ; 0 , 3e^3t] [ e^3t , -2e^3t ; -1 , 5] = [-6e^3t , 30e^3t ; -3e^3t , 15e^3t]

tr(A) = 9e^3t ???????

@&

You neglected one detail in finding psi^-1:

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

so that

A = 1/3 (e^(-3 t) ) * 9 e^(3 t) = 3.

*@

confidence rating #$&*:

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

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): I did not get the traces to equal the same value like I know they should. I think I did the second part of the problem incorrectly, but I'm not sure of what my error was.

------------------------------------------------

Self-critique rating: 3

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

________________________________________

#$&*

&#Good responses. See my notes and let me know if you have questions. &#