ShootingMethod.jl

using DifferentialEquations
using Optim

@public_abstract_type AbstractBVProblem{dType,bType,isinplace,F} <: DEProblem

type BVProblem{dType,bType,initType,F} <: AbstractBVProblem{dType,bType,F}
  f::F
  domin::dType
  bc::bType
  init::initType
end

function BVProblem(f,domin,bc,init=nothing)
  BVProblem{eltype(domin),eltype(bc),eltype(init),typeof(f)}(f,domin,bc,init)
end

function f(t, y, du)
  (x, v) = y
  du[1] = v
  du[2] = -x
end

function solve_(prob::BVProblem)
  bc = prob.bc
  u0 = [bc[1], zero(bc[1])]
  probIt = ODEProblem(prob.f, u0, prob.domin)
  function loss(minimizer)
    probIt.u0[2] = minimizer[1]
    sol = solve(probIt)
    abs(sol[end][1]-bc[2])
  end
  probIt.u0[2] = optimize(loss, [1.], BFGS()).minimizer[1]
  solve(probIt)
end

bc = [0.,1.]
domin = (0.,100.)
init = rand()
bvp = BVProblem(f, domin, bc)
solve_(bvp)