W(t) = W(t_0)*e^(Int(tr[P(s)] ds,t_0,t))
tr[P(s)] = [6 5; -7,-6] = 0, so the Int( tr[P(s)] ds,t_0,t) = 0(t - t_0) = 0
So
W(t) = W(t_0)*e^( Int(tr[P(s)] ds,t_0,t)), plugging in our values
2 = 2*e^(0) = 2*1 = 2, which checks out
?????So this is the process, but not completely sure what’s going on here. Not even sure I got the process completely right. If you can please tell me what I’m missing??????????
confidence rating #$&*:
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Given Solution:
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Self-critique (if necessary):
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Self-critique rating:
@& You got the process.
The trace of the matrix is zero, so the Wronskian is constant (since e^0 = 1).*@
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Question: y ' = A y, with solutions
y_1 = [5; 1]
y_2 = [2 e^(3 t), e^(3 t) ]
Verify that this constitutes a fundamental set.
Find Tr(A).
Show that
psi(t) = [y_1, y_2]
satisfies
psi ' = A * psi
Find A by finding psi ' * psi^-1
Is the result consistent with your result for the trace of A?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
For y_1 = [5; 1] and y_2 = [2 e^(3 t), e^(3 t) ] when we combine we get `psi
`psi(t) = [5, 2e^(3t); 1, e^(3 t)], calculating the determinant
det[`psi(t)] = 3e^(3t),
Determinant was non-zero verifying y_1 and y_2 constitutes a fundamental set.
????????
A=[5, 2e^(3t); 1 , e^(3t)]
Tr[A] = 5+e^(3t)
????????
psi(t) = [y_1, y_2] = [5, 2e^(3t); 1 , e^(3t)]
????I’m confused what the question wants me to do??????
confidence rating #$&*:
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Given Solution:
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Self-critique (if necessary):
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Self-critique rating:
@& psi ' (t) = [ 0, 6 e^(3 t); 0, 3 e^(3 t) ]
W(0) = 3, so
W(t) = 3 e^(3 t) = W(0) e^(3 t).
W(t) = W(0) e^integral(Tr(A) dt) ,
so Tr(A) must be 3.
A psi = [ A y_1, A y_2 ]
where A y_1 and A y_2 are column vectors y_1 'and y_2 '.
psi^-1 = 1 / (3 e^(3 t) * [e^(3 t), -2 e^(3 t); -1, 5 ]
so
A = psi ' psi^-1 = [-2, 10; -1, 5].
You can verify that psi ' found above is equal to A psi for your psi matrix.
*@
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Question:
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
confidence rating #$&*:
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Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
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Self-critique rating:
"
Self-critique (if necessary):
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Self-critique rating:
"
Self-critique (if necessary):
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Self-critique rating:
#*&!
&#Good responses. See my notes and let me know if you have questions. &#