Two Point Boundary Value Problem

I have a non-standard 2 point bvp. The endpoint is not known but needs to be determined as a condition on the forward problem. Then another condition needs to be satisfied at the end point. Presumably one way to do this would be to solve the foward problem step by step until the condition to determine the endpoint is met but this does not seem a very good use of the ode solvers. A worksheet that describes the problem better is attached.