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2.6 General First-Order Linear Difference Equations*
Thus far, we have considered difference equations with constant coefficients.
Now we consider first-order difference equations with arbitrary functions as
coefficients:

y(k + 1) + A(k)y(k) = B(k) (2.30)

The homogeneous equation corresponding to this form satisfies the follow-
ing equation:

l(k + 1) + A(k)l(k) = 0 (2.31)

Its expression can be easily found:

lk( + 1 ) = − Ak lk() () = Ak Ak() ( − 1 ) lk( − 1 ) = … =
 k  (2.32)


l() = 
=−1 ) k+1 Ak Ak( ) ( − 1 )… A()0 0 ∏ [− A i( )]  l()0
(
 i=0 


Assuming that the general solution is of the form:
y(k) = l(k)v(k) (2.33)

let us find v(k). Substituting the above trial solution in the difference equa-
tion, we obtain:

l(k + 1)v(k + 1) + A(k)l(k)v(k) = B(k) (2.34)

Further, assuming that

v(k + 1) = v(k) + ∆v(k) (2.35)

substituting in the difference equation, and recalling that l(k) is the solution
of the homogeneous equation, we obtain:

∆vk () = Bk () (2.36)
lk ( + 1 )

Summing this over the variable k from 0 to k, we deduce that:

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