 Discrete integrating factor

ID: discrete-integrating-factor

For

x_{n + 1} = a_{n} x_{n} + b_{n}

, put

U_{0} = 1

and

U_{n} = \prod_{j = 1}^{n} a_{j}

, with

a_{j} \neq = 0

. Dividing by this accumulated homogeneous growth gives

z_{n} = x_{n} / U_{n - 1}

and

z_{n + 1} - z_{n} = b_{n} / U_{n}

x_{n + 1} = U_{n} (x_{1} + \sum_{j = 1}^{n} \frac{b _{j}}{U _{j}}) .

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