Editing Recurrence relation (section)

== Relationship to differential equations ==
When solving an [[ordinary differential equation]] [[numerical ordinary differential equations|numerically]], one typically encounters a recurrence relation. For example, when solving the [[initial value problem]]

:<math>y'(t) = f(t,y(t)), \ \ y(t_0)=y_0,</math>

with [[Euler's method]] and a step size <math>h</math>, one calculates the values

:<math>y_0=y(t_0), \ \ y_1=y(t_0+h), \ \ y_2=y(t_0+2h), \ \dots</math>

by the recurrence

:<math>\, y_{n+1} = y_n + hf(t_n,y_n), t_n = t_0 + nh </math>

Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the [[discretization]] article.