query 24

#$&*

course Mth 279

8/6 5:06 pm

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: 

 

 Abel's Theorem states

W(t) = W(t_0) e^(integral[trP(s) ds])

Taking the wronskian of solution matrix

W = -5e^(0) + 7e^(0) = 7-5 = 2

Now using Abel's theorem

trace of P(t) = 0

W(-1) = |5e^(1), e^(-1);

-7e^(1), -e^(-1)]

W(-1) = -5e^(0) + 7e^(0) = 2

W(t) = 2 e^(integral[0])

W(t) = 2e^(0) = 2

This agrees with previously calculated Wronskian

 

confidence rating #$&*:

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

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

Given Solution: 

 

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

Self-critique (if necessary):

 

 

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

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: 

 

 verify fundamental set by finding wronskian

W = |5, 2e^(3t);

1, e^(3t)|

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

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

0, 3e^(3t)]

psi'/psi = A

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

1/(e^(3t)) * [-6e^(3t), 30e^(3t); -3e^(3t), 15e^(3t)] = A

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

trA = 0

Abel's theorem

W(t) = W(t_0) e^(integral[trA(s) ds])

3e^(3t) = 3e^(3t) e^(0)

3e^(3t) = 3e^(3t)

This result of A is consistent with trA

confidence rating #$&*:

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

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

Given Solution: 

 

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

Self-critique (if necessary):

 

 

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

Self-critique rating:

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

Question: 

 

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 

"

Self-critique (if necessary):

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

Self-critique rating:

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

Question: 

 

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 

"

Self-critique (if necessary):

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

Self-critique rating:

#*&!

&#This looks good. Let me know if you have any questions. &#