There is no direct mechanism to incorporate integral constraints using BVP4C. A work around is to convert the integral constraints into differential constraints by introducing additional states. For example, consider the following problem:
Solve: x_1’ = x_2x_2’ = 1 Subject to the boundary constraints : x_1(0)=0 [integral(x_2^2) from 0 to 1] = 5.
To solve this problem, introduce an additional state x_3 such that, x_3’ = x_2^2 with boundary conditions: x_3(0)=0 x_3(1)=5.
Now you can use BVP4C to solve the following system of 3 ODEs with 3 boundary conditions: x_1’ = x_2 x_2’ = 1 x_3’=x_2^2 x_1(0)=0 x_3(0)=0 x_3(1)=5
