Query 15

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course mth 279

7/16 9:38pm

Query 15 Differential Equations*********************************************

Question:  Suppose y1 and y2 are solutions to y '' + 2 t y ' + t^2 y = 0.  If y1(3) = 0, y1 ' (3) = 0, y2(3) = 1 and y2 ' (3) = 2, can you say whether {y1, y2} is a fundamental set?  If so, is it or isn't it?

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Your solution: 

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I'm not sure how to solve this problem.

I know that to be a fundamental set the wronskian cannot equal zero. We aren't given the function y1 and y2, so I'm not sure how to compute the wronskian. I think that we might be able to reverse engineer the functions using the given initial conditions, but I don't know how.

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You know the values of the necessary functions for t = 3.

If the Wronskian is nonzero at any point within the domain, it's nonzero at all points within the domain.

So you can verify by using just the values at t = 3.

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Given Solution: 

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Self-critique (if necessary):

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Question:  Are y1 = 2 e^(-2 t) cos(t) and y2 = e^(-2 t) sin(t) solutions to the equation

y '' + 4 y ' + 5 y = 0?

What are the initial conditions at t = 0?

Is {y1, y2} a fundamental set?

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Your solution: 

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find initial conditions at t = 0 by plugging in y1 and y1'

we find y(0) = 2

and y'(0) = 6

W(t) = (2 e^(-2t)*cos(t) )* (e^(-2t) * (cos(t) - 2 sin(t)) - ( e^(-2t) ( -4 cos(t) - 2 sin(t) ) * (e ^(-2t) sin (t) ) = 2 e^(-4t)

not equal to zero, thus y1 and y2 are the fundamental set.

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confidence rating #$&*:232; 

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Given Solution: 

 

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Question:  y1_bar = 2 y1 - 2 y2 and y2_bar = y1 - y2.  Is {y1_bar, y2_bar} a fundamental set?

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Your solution: 

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 I have no idea what this is asking. I found something similar in the book (1st edition) on page 141, but I don't understand it there either.

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Evaluate the determinant of the matrix

[y1_bar, y2_bar; y1_bar ', y2_bar ' ] =

[ 2 y1 - y2 y1 - y2; 2 y1' - y2 ', y1' - y2'].

Simplify your result.

If the result simplifies to 0, then this is not a fundamental set.

If it does not, and if y1 and y2 are assumed to constitute a fundamental set, then you might be able to conclude that these two functions do constitute a fundamental set.

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Given Solution:  Note that y_1_bar = 2 * y_2_bar. 

 

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Question:  Is {e^t, 2 e^(-t), sinh (t) } a fundamental set on the interval (-infinity, infinity)?

 

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Your solution: 

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W(t) = 2 e^(-t) * (e^(2t) + 2 e^t sin(t) - 1 )

not equal to zero so yes, it is a fundamental set.

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 Confidence rating:  

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Given Solution: 

 

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Self-critique (if necessary):

  

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sinh(t) = (e^t - e^(-t) ) / 2, which is a linear combination of the first two function (of the form (y1 + 1/2 y2) / 2).

The Wronskian should not be zero.

You should re-evaluate the Wronksian.

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