Query 26

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course MTH 279

2:08 pm 5/4

Query 26 Differential Equations*********************************************

Question:  Find the solutions to y ' = A y when

A = [ 5, 3; -4, -3 ]

and

y(1) = [2; 0].

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Your solution: 

lamba_1 = -1

lambda_2 = 3

x_1 = [1,-2]

x_2 = [3, -2]

y = c1*e^-t*[1,-2] + c2*e^3t*[3,-2]

c1*e^-1 + c2*e^3*3=2

c1*e^-1*-2+ c2*e^3*-2 = 0

c1= -e

c2 = e^-3

y = -e*e^-t*[1,-2] + e^(-3)*e^3t*[3,-2]

 

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:  Find the solutions to y ' = A y when

A = [ 4,2,0; 0,1,3; 0,0, -2 ]

and

y(0) = [-1;0;3].

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Your solution: 

lamba_1 = 4

lamba_2 = -2

lambd_3 = 1

x_1 = [1,0,0]

x_2 = [1, -3, 3]

x_3 = [2, -3, 0]

y = c1e^4t*[1,0,0] + c2e^(-2t)*[1, -3, 3]+ c3e^t*[2, -3, 0]

three eqs for c1 c2 & c3

c1 = 0

c2 =1

c3 = -1

y = e^(-2t)*[1, -3, 3]-e^t*[2, -3, 0]

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:  Three tanks, each full of a solution and all of equal volume V, are each connected to each of the others by two pipes.  Each tank also has a third pipe through which pure water flows into it, and a fourth through which water exits.

The flow rate r through every pipe is the same.

Write the system of equations that relates the quantities Q_1, Q_2 and Q_3 representing the amount of solute in each tank as a function of time.

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Your solution: 

Q/V*r = solute thru pipe where r = rate and Q = amt of solute and V = volume

Q1' = -Q1/V*r - Q1/V*r + Q2/V*r + Q3/V*r

Q2' = -Q2/V*r - Q2/V*r + Q3/V*r + Q1/V*r

Q3' = -Q3/V*r - Q3/V*r + Q2/V*r + Q1/V*r 

 

 Confidence rating:

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Given Solution: 

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Self-critique (if necessary):

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&#Very good work. Let me know if you have questions. &#