search
Log In
0 votes
135 views

What is the general form of the particular solution guaranteed to exist of the linear nonhomogeneous recurrence relation
$a_n$=$6a_{n-1}$-$12a_{n-2}$+$8a_{n-3}$+F(n) if

  1. F(n)=$n^2$
  2. F(n)=$2^n$
  3. F(n)=$n2^n$
  4. F(n)=$(-2)^n$
  5. F(n)=$n^22^n$
  6. F(n)=$n^3(-2)^n$
  7. F(n)=3
     
in Combinatory 135 views
0
how to find the particular solution for c, e, f?
0
First solve the equation $\alpha^{3} - 6\alpha ^{2} +12\alpha -8=0$

So, here $\alpha=2,2,2$

Now, for particular solutions,

For $c)$ root $2$ has multiplicity $m=3$

So,

$a_{n}^{(p)} = n^{3}(P_{1}n + P_{0})2^{n}$

Put it in recurrence and find $P_1$ and $P_0$

For $e)$

$a_{n}^{(p)} = n^{3}(P_2n^2+P_{1}n + P_{0})2^{n}$

Put it in recurrence and find $P_2,P_1$ and $P_0$

For $f)$

$a_{n}^{(p)} = (P_3n^3+P_2n^2+P_{1}n + P_{0})(-2)^{n}$

Put it in recurrence and find $P_3,P_2,P_1$ and $P_0$

Please log in or register to answer this question.

Related questions

0 votes
0 answers
1
89 views
Consider the nonhomogeneous linear recurrence relation $a_n$=$3a_{n-1}$+$2^n$ in the book solution is given $a_n$=$-2^{n+1}$ but I’m getting $a_n$=$3^{n+1}-2^{n+1}$
asked May 13, 2019 in Combinatory aditi19 89 views
0 votes
1 answer
2
152 views
How many bit sequences of length seven contain an even number of 0s? I'm trying to solve this using recurrence relation Is my approach correct? Let T(n) be the string having even number of 0s T(1)=1 {1} T(2)=2 {00, 11} T(3)=4 {001, 010, 100, 111} Case 1-we add 1 to strings of length ... =T(n-1) Case 2-we add 0 to strings of length n-1 having odd number of 0s T(n)=T(n-1) Hence, we have T(n)=2T(n-1)
asked Apr 29, 2019 in Combinatory aditi19 152 views
0 votes
2 answers
3
0 votes
0 answers
4
12 views
What is the general form of the particular solution guaranteed to exist by Theorem $6$ of the linear nonhomogeneous recurrence relation $a_{n} = 6a_{n-1} - 12a_{n-2} + 8a_{n-3} + F (n)$ if $F (n) = n^{2}?$ $F (n) = 2^{n}?$ $F (n) = n2^{n}?$ $F (n) = (-2)^{n}?$ $F (n) = n^{2}2^{n}?$ $F (n) = n^{3}(-2)^{n}?$ $F (n) = 3?$
asked May 5 in Combinatory Lakshman Patel RJIT 12 views
...