15.5. Example with Degree 2 Characteristic Equations

Try these first

1. 1.

   What is the characteristic equation of the recurrence relation f(n) = 6f(n-1) + 9f(n-2)?

   Hint

   Write the characteristic equation based on the recurrence relation.

2. 2.

   Define what initial conditions are in the context of recurrence relations.

   Hint

   Think about how initial values are used in calculations.

3. 3.

   What happens to the general solution if the roots of the characteristic equation are repeated?

   • It remains unchanged.
   • It becomes a polynomial multiplied by the root raised to its multiplicity.
   • It is impossible to determine
   • It simplifies to a linear function.

   Hint

   Recall the form of solutions for distinct versus repeated roots.

4. 4.

   True or False: Initial conditions can be ignored when finding general solutions from characteristic equations.

   • True
   • False

   Hint

   Consider the role of initial conditions in solving recurrence relations.

5. 5.

   Given the recurrence relation f(n) = 2f(n-1) + f(n-2) with initial conditions f(0)=1 and f(1)=2, find the general solution.

   Hint

   Don't forget to check the nature of the roots before proceeding with initial conditions.

6. 6.

   Determine the general form of the recurrence relation defined by f(n) = 4f(n-1) - 4f(n-2) with no initial conditions.

   Hint

   Focus on the given recurrence structure to create the characteristic equation first.