Shooting Method

y(0)

-1.04

y'(0)

0

Consider the boundary-value problem

y''=-xy

with

y(0)=1

and

y(4)=1/2

. You can solve this using only the initial condition by varying the initial slope until the function

y(x)

satisfies the condition at the endpoint (that is, the solution goes through the orange dot at

(4,1/2)

). This is called the shooting method. For ten years, Mathematica has used a sophisticated version of this for boundary-value problems in NDSolve.