Math Problem Statement
Solve the recurrence relation 𝑎 − 7𝑎ିଵ + 12𝑎ିଶ = 𝑟. 4 .
Solution
We are tasked with solving the recurrence relation:
where .
Step 1: Solve the Homogeneous Equation
The first step is to solve the homogeneous part of the recurrence relation:
Assume a solution of the form . Substituting into the homogeneous equation:
Dividing through by gives the characteristic equation:
We solve this quadratic equation:
So, the roots are:
Thus, the general solution to the homogeneous equation is:
Step 2: Solve the Non-Homogeneous Equation
Now, we solve the non-homogeneous equation:
We need to find a particular solution, , to this equation. Since the right-hand side is linear in , we try a particular solution of the form:
Substitute this into the recurrence relation:
Simplifying both sides:
Equating the coefficients of and the constant terms:
- , which gives
- , which gives , or , so
Thus, the particular solution is:
Step 3: General Solution
The general solution to the recurrence relation is the sum of the homogeneous and particular solutions:
Conclusion
The solution to the recurrence relation is:
To find the constants and , you would need initial conditions (values for , , etc.).
Do you have any questions or would you like further details? Here are some questions that expand on the topic:
- How do we handle initial conditions in recurrence relations?
- Can we apply the same method for non-homogeneous terms of higher degree?
- What happens if the characteristic equation has repeated roots?
- How does this method apply to second-order linear recurrence relations in general?
- How would you solve a recurrence relation using matrix methods?
Tip: Always check the form of the non-homogeneous term before assuming a particular solution!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Recurrence Relations
Linear Homogeneous Equations
Characteristic Equation
Non-Homogeneous Solutions
Formulas
Characteristic equation x^2 - 7x + 12 = 0
General solution of a linear recurrence relation: a_r = C_1 4^r + C_2 3^r + particular solution
Theorems
Method of Solving Linear Recurrence Relations
Homogeneous and Non-Homogeneous Linear Recurrence Theorem
Suitable Grade Level
Grades 11-12 or early undergraduate level
Related Recommendation
Solving Recurrence Relation a_r - 5a_{r-1} + 6a_{r-2} = 2 with Initial Conditions a_0 = 1, a_1 = 1
Solve Recurrence Relation a_r − 7a_{r−1} + 12a_{r−2} = r
Solving Recurrence Relation a_r = -2a_(r-1) - a_(r-2) + 5
Recurrence Relation Solution for a_n = 2a_{n-1} + 3 - 6a_{n-3} with Initial Conditions
Solve Recurrence Relation a_n = 7a_(n-1) - 10a_(n-2) with Initial Condition a_0 = 2