solve the difference equation x_{n+2} - 4x_{n}= n

Question

solve the difference equation x_{n+2} - 4x_{n}= n

 

with initial conditions: x_{0} = 0, x_{1} = 1/3

Answer 1

x_o = 0

x₁ = 1/3

 

when n = 0, & using the formula x_n+2 = n + 4x_n

we get x₂ = 0 + 4(0) so x₂ = 0.

 

when n = 1, x₃ = 1 + 4(1/3) = 7/3.

 

when n = 2, x₄ = 2 + 4(0) = 2.

 

when n = 3, x₅ = 3 + 4(7/3) = 37/3.

 

A peculiar sequence (because of the peculiar recursion rule).