Distinct Roots

If the characteristic roots of Eq. (1.5) are distinct, then a fundamental set of solutions is

yk = k i = 1, 2,…, n

and the general solution of the homogeneous equation is

yk = c1rk + c2rk + ••• + cnrt (1.7)

where cx, c2,…, cn are n arbitrary constants. example. Find the general solution of

yk+3 – 7yk+2 + 14yk+1 – 8yk = °-

Let yk = crk. Substitution into the difference equation yields the characteristic equation

r3 – 7r2 + 14r – 8 = 0

or

(r – 1) (r – 2) (r – 4) = 0.

The characteristic roots are r = 1, 2, and 4. Therefore, the general solution is

yk = A + B2k + C4k,

where A, B, and C are arbitrary constants.

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