University of Florida/Egm4313/s12.team8.dupre/R2.3

R2.3

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Problem Statement

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Find a general solution. Check your answer by substitution.

a)   (3-1)

b)  (3-2)

Solution

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The quadratic formula is necessary for these solutions:

 

Part a

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Plugging into the quadratic formula:

 

This shows us that the roots of the equation are:

 

Therefore, the general equation is:

      (3-3)

Substitution
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We need to first find the first and second derivatives of equation (3-3):

 

 

Plugging into equation (3-1), we find:

  (3-4)

Continuing to solve:

  (3-5)

This shows that the general equation is correct, since everything cancels out to 0.

Part b

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Plugging into the quadratic formula:

 

The roots are, therefore:

 

Therefore, the general solution to (3-2) is:

  (3-6)

Substitution
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We must first find the first and second derivatives of equation (3-6):

 

 

Plugging into equation (3-2):

 

 

Finally, plugging (3-6) and it's first and second derivatives into equation (3-2), we find:

 

 

 

Since this equals 0, we know that the general equation (3-6) is correct.