7.7.1. Case 1: Repeated Roots and Duplication

Try these first

1. 1.

   What should you do if the trial solution overlaps with the complementary function?

   Hint

   Think about how many times the term appears in the complementary function.

2. 2.

   Define repeated roots.

   Hint

   Consider what happens to solutions when roots aren't distinct.

3. 3.

   What is the purpose of modifying a trial function in the method of undetermined coefficients?

   • To ensure duplication is eliminated.
   • To make calculations easier.
   • To change the forcing function.
   • It serves no particular purpose.

   Hint

   Consider the role of the complementary function.

4. 4.

   True or False: Duplication can occur only with polynomial functions.

   • True
   • False

   Hint

   Think about other functional forms like exponentials.

5. 5.

   Given the differential equation y'' - 6y' + 9y = 12e^(3x), identify the trial solution and show your work.

   Hint

   Evaluate the characteristic equation carefully.

6. 6.

   Solve for y in the equation y'' + 4y = 2sin(2x) and adjust the trial solution accordingly.

   Hint

   Determine the complementary function first.