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

From Wikiversity
Jump to navigation Jump to search

R2.3[edit]

Problem Statement[edit]

Find a general solution. Check your answer by substitution.

a) (3-1)

b) (3-2)

Solution[edit]

The quadratic formula is necessary for these solutions:

Part a[edit]

Plugging into the quadratic formula:



This shows us that the roots of the equation are:



Therefore, the general equation is:

     (3-3)

Substitution[edit]

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[edit]

Plugging into the quadratic formula:



The roots are, therefore:



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

(3-6)

Substitution[edit]

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.