simply support with middle support nonlinear beam bending solving using the shooting method

I am trying to solve for the deflection along the length of a straight beam. I plan to use the shooting method since the elasticity modulus changes along the length of the beam. The beam is made up of pin supports on each end, and a roller support 1/3 of the way down the beam. My code works for a simply supported beam, but i cannot figure how to get my code to work correctly when trying to solve it with the roller support. How can I go about solving this problem?