How can ı solve this difference equation ?

Consider the difference equation:

x(k + 2) - 1.3679x(k + 1) + 0.3679x(k) = 0.3679u(k + 1) + 0.2642u(k) where x(k) = 0 for k ≤0. The input u(k) is given by

u(k) = 0, k<0

u(0) = 1.5820

u(1) = -0.5820

u(k) = 0, k = 2,3,4,...

Determine the output x(k). Solve this problem both analytically and computationally with MATLAB

Answer 1

[Torsten]

um2 = 0;
um1 = 0;
u0 = 1.5820;
u1 = -0.5820;
u2 = 0;
xm2 = 0;
xm1 = 0;
x0 = 1.3679*xm1 - 0.3679*xm2 + 0.3679*um1 + 0.2642*um2

x0 = 0

x1 = 1.3679*x0 - 0.3679*xm1 + 0.3679*u0 + 0.2642*um1

x1 = 0.5820

x2 = 1.3679*x1 - 0.3679*x0 + 0.3679*u1 + 0.2642*u0

x2 = 1.0000

x3 = 1.3679*x2 - 0.3679*x1 + 0.3679*u2 + 0.2642*u1

x3 = 1.0000

So you have x2 and x3 and the formula for xk (k>=2) reads

x(k+2) - 1.3679*x(k+1) + 0.3679*x(k) = 0

Can you take it from here ?

[FURKAN]

Yes thank you