Answered: (Note: if the math equations and the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(Note: if the math equations and the text below are overlapping, click twice on the piece of math that is causing the issue. That should fix it.)

f₀=−3,

f₁=−29,

8f_n=f_n+1+15f_n-1

The following four questions are all based on the paragraph above. You have to answer all of them in order to be able to click the submit button down below.

Here, r₁and r₂are the roots of the characteristic equation of the recurrence relation and a1 and a2 are two constants such that fn=a₁r₁ⁿ+a₂r₂ⁿsatisfy the recurrence.

(1)f₂=?

(2) 3(r₁+r₂)=?

3) a₁r₂+a₂r₁=?

(4) f₁₁=?

Expert Solution

Step 1

Consider the recurrence relation $a_{n} = b a_{n - 1} + c a_{n - 2}$ .

Let the characteristic equation of the recurrence relation be $a r^{2} + b r + c = 0$ , where a, b and c are constants.

To find the characteristic roots, use the quadratic formula $r = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

The general solution of the recurrence relation is as follows.

If the characteristic roots r₁ and r₂ are distinct real roots, then the general solution is $a_{n} = a_{1} {r_{1}}^{n} + a_{2} {r_{2}}^{n}$ .
If the characteristic root r₀ has multiplicity 2, then the general solution is $a_{n} = (a_{1} + a_{2} n) {r_{0}}^{n}$ .