19.2.4. Methods for Finding Particular Solutions

Try these first

1. 1.

   Define what a non-homogeneous recurrence equation is?

   Hint

   Think of how it differs from a homogeneous one.

2. 2.

   What is the associated homogeneous relation?

   Hint

   Recall how we simplify the equation.

3. 3.

   What is the first step to solve a linear non-homogeneous recurrence equation?

   • Find the roots of F(n)
   • Form the associated homogeneous relation
   • Guess the particular solution

   Hint

   Think about what we do to simplify the original equation.

4. 4.

   True or False: The particular solution satisfies only the homogeneous part of the recurrence.

   • True
   • False

   Hint

   Recall the components of a particular solution.

5. 5.

   Given the recurrence relation a_n = 4a_(n-1) + 2n, derive the general solution.

   Hint

   Remember the steps of chopping off and combining solutions.

6. 6.

   If the recurrence is defined as a_n = 5a_(n-1) + 7, determine the structure of your particular solution based on the roots identified.

   Hint

   Think about what it means when the constant is not a root of the characteristic equation.