In the MATLAB exists the command filter that permits the resolution of difference equation numerically.
An important difference equation that express a link between input ad output in a LTI is
\[\sum_{k=0}^N a_ky(n-k) = \sum_{m=0}^M b_mx(n-m)\]
this equation can be solved using filter(b,a,x) where $b=[b_0 ... b_m],\; a=[a_0 ... a_k]$.
\[z(n) = conv(x,y) = x(n)*y(n) = \sum_{k=-\infty}^{+\infty} x(k)y(n-k)\]
Considering two sequence with finite length we can observe that:
\[filter(x,1,y)\Rightarrow 1\cdot z(n) =\sum_{m=0}^M x(m)y(n-m)\]
from below expression we can see that using filter function we can compute convolution of x and y but the result will have the dimension of x
