Linear Algebra? (Or Differential Equation?)

[mudkip123]

Find u_n, v_n, and t_n in terms of n for the following system:
{
u_n+1 = -u_n + 2v_n + t_n
v_n+1 = v_n - t_n
t_n+1 = 2t_n

For u₀, v₀, t₀ given


This isn't homework, it's on a study guide for my midterm.

 

[tiny-tim]

welcome to pf!

hi mudkip123! welcome to pf!

you're looking for a combination a_n = Au_n + Bv_n + Ct_n such that a_n+1 = Da_n

 

[Ray Vickson]

Find u_n, v_n, and t_n in terms of n for the following system:
{
u_n+1 = -u_n + 2v_n + t_n
v_n+1 = v_n - t_n
t_n+1 = 2t_n

For u₀, v₀, t₀ given


This isn't homework, it's on a study guide for my midterm.

First solve the third one $t_{n+1} = 2 t_n,$ to get $t_n = t_0 2^{n-1}.$ Now look at the second one $v_{n+1} = v_n -t_n.$ It gives $v_n = v_0 -\sum_{j=0}^{n-1} t_j,$ which is computable. Now you have $u_{n+1} = -u_n + f(n)$ with a known function f(n), so you can solve it easily: $$u_1 = f(0)-u_0, \; u_2 = f(1) - u_1 = f(1)-f(0) + u_0, \; u_3 = f(2) - u_2 = f(2)-f(1)+f(0) - u_0,$$ etc.

RGV

 

[mudkip123]

This isn't in my linear algebra textbook so I don't know how to do this...

 

[micromass]

You have been given enough hints. We will only help you further if you make an attempt now.