Partikularni reseni rekurence

asked 2014-11-30 10:46:51 +0100

kuzmamar
23 ●2 ●2 ●4

updated 2014-11-30 16:29:27 +0100

Tomáš Kalvoda
960 ●10 ●16 ●24

Ahoj, jak se prosim vas urci partikularni reseni u teto rekurence?

$$ a_n = a_{n-1} + 2n, \quad a_1 = 1 $$

nejdrive spocitam pridruzenou homogenni rovnici:

$$ a_n - a_{n-1} = 0 $$

char. polynom je:

$$ p(r) = r - 1 $$

reseni polynomu:

$$ r - 1 = 0 \quad \Rightarrow \quad r = 1 $$

takze homogenni cast reseni je: $ A \cdot 1^n = A $

Jak najit partikularni reseni?

Zkousel jsem:

$$ f(n) = 2n $$

$$ b(n) = B\cdot n $$

po dosazeni do puvodni rekurence:

$$ Bn = B(n-1) + 2n $$

$$ Bn = Bn - B + 2n $$

$$ 0 = -B + 2n $$ $$ B = 2n $$ ted nevim jak dal :D

Diky za rady