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Solve Finite Difference Equation

See attached file for full problem description with equation.

Find analytically the solution of this difference equation with the given initial values:

Without computing the solution recursively, predict whether such a computation would be stable.

(Note: A numerical process is unstable if small errors made at one stage of the process are magnified in subsequent stages and seriously degrade the accuracy of the overall calculation.)


Solution Preview

Please see the attached file.

Difference equation:
x_{n+1} = -0.2 x_n + 0.99 x_n. eqno(1)
Starting conditions:
x_0 = 1, hskip 1cm x_1=0.9. eqno(2)
This is a linear homogeneous equation with constant coefficients. Therefore its solutions are a linear combination of basic solutions of form
B_n = cq^n,eqno(3)
where $c$ and $q$ are constants.
Substituting the basic form (3) into equation ...