Snapshot 1: the approximations approach the true solution with increasing iterations of Picard's method.

Snapshot 2: the approximation after the first iteration already captures the behavior of the solution.

Snapshot 3: although the initial guess is poor, the approximations rapidly improve.

Picard's method approximates the solution

to a first order ordinary differential equation of the form

,

with initial condition

. The solution is

.

Picard's method uses an initial guess

to generate succesive approximations to the solution as

such that after the

iteration

.

Above, we take

, with at

,

. Several choices for the initial guess

and differential equation

are possible. After each iteration, the mean square error of the approximation is computed by sampling the true solution (in blue) and the approximation

at evenly spaced points in

.