2.9 First-Order Difference Equations Definition
y n + 1 = f ( n , y n ) , n = 0 , 1 , 2... y_{n+1} = f(n ,y_n) ,\quad n = 0 ,1 ,2... y n + 1 = f ( n , y n ) , n = 0 , 1 , 2... It is called first-order difference equation. if f is a function of y n y_n y n y 0 , y 1 , y 2 y_0 ,y_1 ,y_2 y 0 , y 1 , y 2
General case
y n + 1 = f ( y n ) , n = 0 , 1 , 2... y_{n+1} = f(y_n), \quad n = 0 ,1, 2... y n + 1 = f ( y n ) , n = 0 , 1 , 2... theny 1 = f ( y 0 ) , y 2 = f ( y 1 ) = f ( f ( y 0 ) = f 2 ( y 0 ) y_1 = f(y_0) ,\quad y_2 = f(y_1) = f(f(y_0) = f^2(y_0) y 1 = f ( y 0 ) , y 2 = f ( y 1 ) = f ( f ( y 0 ) = f 2 ( y 0 )
In general , the n t h n^{th} n t h y n y_n y n y n = f n ( y 0 ) y_n = f^n(y_0) y n = f n ( y 0 )
Linear Equations
Case-1
y n + 1 = ρ n y n , n = 0 , 1 , 2... y_{n+1} = \rho_ny_n,\quad n= 0 ,1 ,2... y n + 1 = ρ n y n , n = 0 , 1 , 2... In general ,y n = ρ n − 1 . . . ρ 0 y 0 , n = 1 , 2 , . . . y_n = \rho_{n-1}...\rho_0y_0 ,\quad n = 1 ,2 ,... y n = ρ n − 1 ... ρ 0 y 0 , n = 1 , 2 , ...
if ρ n = ρ ∀ n ,the equations becomes y n = ρ n y 0 \rho_n = \rho \ \ \forall n \text{ ,the equations becomes } y_n = \rho^n y_0 ρ n = ρ ∀ n ,the equations becomes y n = ρ n y 0
lim n → ∞ y n = { 0 , i f ∣ ρ ∣ < 1 ; y 0 , i f ρ = 1 ; doesn’t exist , otherwise. \lim_{n \to \infty}{y_n} = \begin{cases} 0, \quad if \ |\rho| < 1; \\ \quad \\ y_0, \quad if \ \rho = 1; \\ \\ \text{doesn't exist , otherwise.} \end{cases} n → ∞ lim y n = ⎩ ⎨ ⎧ 0 , i f ∣ ρ ∣ < 1 ; y 0 , i f ρ = 1 ; doesn’t exist , otherwise. Case-2
y n + 1 = ρ y n + b n , n = 0 , 1 , 2... y_{n+1} = \rho y_n + b_n,\quad n = 0,1,2 ... y n + 1 = ρ y n + b n , n = 0 , 1 , 2... y 1 = ρ y 0 + b 0 y_1 = \rho y_0 + b_0 y 1 = ρ y 0 + b 0 y 2 = ρ ( ρ y 0 + b 0 ) + b 1 = ρ 2 y 0 + ρ b 0 + b 1 y_2 = \rho(\rho y_0 + b_0) + b_1 = \rho^2 y_0 + \rho b_0 + b_1 y 2 = ρ ( ρ y 0 + b 0 ) + b 1 = ρ 2 y 0 + ρ b 0 + b 1 y n = ρ n y 0 + ∑ j = 0 n − 1 ρ n − 1 − j b j y_n = \rho^n y_0 + \sum_{j = 0}^{n-1} \rho^{n-1-j} b_j y n = ρ n y 0 + j = 0 ∑ n − 1 ρ n − 1 − j b j In special case where b n = b ≠ 0 for all n ,the difference equations is \text{In special case where }b_n = b \neq 0\text{for all n ,the difference equations is } In special case where b n = b = 0 for all n ,the difference equations is
y n + 1 = ρ y n + b y_{n+1} = \rho y_n + b y n + 1 = ρ y n + b If ρ ≠ 1 , we can write the solution in the more compact form \text{If }\rho \neq 1, \text{we can write the solution in the more compact form} If ρ = 1 , we can write the solution in the more compact form
y n = ρ n ( y 0 − b 1 − ρ ) + b 1 − ρ y_n = \rho^n (y_0 - \frac{b}{1-\rho})+\frac{b}{1 - \rho} y n = ρ n ( y 0 − 1 − ρ b ) + 1 − ρ b