BVProblem MethodError

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Following the documentation at Boundary Value Problems · DifferentialEquations.jl your implementation should probably look like

using DifferentialEquations

E = 200e9     # Pa
A = (0.01)^2  # m^2

L = 2         # m   length beam
b = 1000      # N/m
P = 250       # N

EE(x) = E
AA(x) = A
bb(x) = b

function diffeq!(dy,y,p,x)
    dy[1] = y[2]/(EE(x)*AA(x))
    dy[2] = -bb(x)
end

function bc!(residual,y,p,x)
   residual[1] = y[1][1] - 0
   residual[2] = y[end][2] - L
end

bvp = BVProblem(diffeq!, bc!, [0,0], (0.0,L))
sol = solve(bvp, GeneralMIRK4(), dt=0.05)
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