3. Solve the recurrence relation. Given: • a - 3 a = 6 a = 6a+ 7a_2 Show your work in the space provided. a Find c, and c, (6)a_, + (7)ª,-2 b. Substitute c, and c, into the following equation: - cr -, = 0 Show your work here:
3. Solve the recurrence relation. Given: • a - 3 a = 6 a = 6a+ 7a_2 Show your work in the space provided. a Find c, and c, (6)a_, + (7)ª,-2 b. Substitute c, and c, into the following equation: - cr -, = 0 Show your work here:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Solve the recurrence relation. Given:
- \( a_0 = 3 \)
- \( a_1 = 6 \)
- \( a_n = 6a_{n-1} + 7a_{n-2} \)
Show your work in the space provided.
a. Find \( c_1 \) and \( c_2 \):
\[
a_n = (6)a_{n-1} + (7)a_{n-2}
\]
\[
c_1 =
\]
\[
c_2 =
\]
b. Substitute \( c_1 \) and \( c_2 \) into the following equation:
\[
r^2 - c_1r - c_2 = 0
\]
Show your work here:
[Space is provided for showing work.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17474514-9340-40f1-94d8-ebb350c2aeb9%2Fc501f336-97f6-49bf-9ce2-b756dbbfd47c%2Fomnb5mp_processed.png&w=3840&q=75)
Transcribed Image Text:3. Solve the recurrence relation. Given:
- \( a_0 = 3 \)
- \( a_1 = 6 \)
- \( a_n = 6a_{n-1} + 7a_{n-2} \)
Show your work in the space provided.
a. Find \( c_1 \) and \( c_2 \):
\[
a_n = (6)a_{n-1} + (7)a_{n-2}
\]
\[
c_1 =
\]
\[
c_2 =
\]
b. Substitute \( c_1 \) and \( c_2 \) into the following equation:
\[
r^2 - c_1r - c_2 = 0
\]
Show your work here:
[Space is provided for showing work.]
![### Quadratic Equation Exercise
**Part c: Identify coefficients in the quadratic equation.**
- \(a =\)
- \(b =\)
- \(c =\)
**Part d: Quadratic Formula to Find Roots**
Use the quadratic formula to find the two roots. Here is the quadratic formula:
\[
\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
**Show your work here:**
\[
\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} =
\]
**Part e: Substitute Roots into the Equation**
Substitute two roots, \(r_1\) and \(r_2\), into the equation:
\[
a_n = \alpha r_1^n + \alpha r_2^n
\]
**Show your work here:**
---
This exercise guides students through the process of identifying the coefficients in a quadratic equation, using the quadratic formula to find the roots, and then substituting these roots into another given equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17474514-9340-40f1-94d8-ebb350c2aeb9%2Fc501f336-97f6-49bf-9ce2-b756dbbfd47c%2F9xkime_processed.png&w=3840&q=75)
Transcribed Image Text:### Quadratic Equation Exercise
**Part c: Identify coefficients in the quadratic equation.**
- \(a =\)
- \(b =\)
- \(c =\)
**Part d: Quadratic Formula to Find Roots**
Use the quadratic formula to find the two roots. Here is the quadratic formula:
\[
\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
**Show your work here:**
\[
\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} =
\]
**Part e: Substitute Roots into the Equation**
Substitute two roots, \(r_1\) and \(r_2\), into the equation:
\[
a_n = \alpha r_1^n + \alpha r_2^n
\]
**Show your work here:**
---
This exercise guides students through the process of identifying the coefficients in a quadratic equation, using the quadratic formula to find the roots, and then substituting these roots into another given equation.
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