Question

For each recurrence relation and initial conditions, find: (i) general solution;

(ii) unique solution with the given initial conditions:

(a) an = 3an−1 + 10an−2; a0 = 5, a1 = 11

Answer 1

$$a_n = 3a_{n−1} + 10a_{n−2}$$

$$a_0=5$$

$$a_1=11$$

(i)

Characteristic equation: $r^2-3r-10=0$

$$(r+2)(r-5)=0$$

Characteristic roots: $r_1=-2, r_2=5$

The general solution is

$$a_n=\alpha_1(-2)^n+\alpha_2(5)^n$$

for some constants $\alpha_1$ and $\alpha_2.$

ii) Find the unique solution with the given initial conditions

$$a_0=\alpha_1+\alpha_2=5$$

$$a_1=\alpha_1(-2)+\alpha_2(5)=11$$

$$\alpha_1+\alpha_2=5$$

$$7\alpha_2=21$$

$$\alpha_1=2, \alpha_2=3$$

The unique solution with the given initial conditions is

$$a_n=2(-2)^n+3(5)^n$$