Query 14

#$&*

course MTH 279

6:10 pm 3/31 Seems query 13 was a repeat of some of query 12's questions

Query 14 Differential Equations*********************************************

Question:  Decide whether y_1 = 3 e^t, y_2 = e^(t + 3) are solutions to the equation y '' - y = 0.  If so determine whether the two solutions are linearly independent.  If the solutions are linearly independent then find the general solution, as well as a particular solution for which y (-1) = 1 and y ' (-1) = 0.

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Your solution: 

Simple substitution confirms these are both solutions but the Wronskian is:

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

this means they aren't linearly independent and in fact

x(3e^t) = e^(t+3) where x = 1/3*e^3

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:  Decide whether y_1 = e^(-t) and y_2 = 2 e^(1 - t) are solutions to the equation y '' + 2 y ' + y = 0.  If so determine whether the two solutions are linearly independent.  If the solutions are linearly independent then find the general solution, as well as a particular solution for which y (0) = 1 and y ' (0) = 0.

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Your solution: 

 Substitution confirms these are solutions

y= e^-t, y'=-e^-t, y''= e^-t

y= 2e^(1-t), y'= -2e^(1-t), y''= 2e^(1-t)

W= e^-t*(-2e^(1-t))-(-e^-t)*2e^(1-t) = 0

So they're not lin ind and

2e^1 *(e^(-t))= 2e^(1-t)

confidence rating #$&*:8232;Given Solution: 

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Self-critique (if necessary):

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Self-critique rating:

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Question:  Suppose y_1 and y_2 are solutions to the equation

y '' + alpha y ' + beta y = 0

and that y_1 = e^(2 t).  Suppose also that the Wronskian is e^(-t).

What are the values of alpha and beta?

 

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Your solution: 

y1= e^2t, y1'= 2e^2t, y1''=4e^2t

4e^2t+alpha*2e^2t+beta*e^2t =0

4+2alpha+beta= 0

W= e^-t = e^2t * y2' - y2*2e^2t

y'2-2y2= e^-3t

So y2 must be multiple of e^-3t

Ae^-3t - 2Ae^-3t = e^-3t

A = -1

y2 = -e^-3t, y2'= 3e^-3t, y2''= -9e^-3t

-9+3alpha-beta =0

4+2alpha+beta= 0

alpha = 1, beta =-6

confidence rating #$&*:

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Given Solution: 

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Self-critique (if necessary):

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04-01-2011