I met this kind of BVP. The ODEs are like: $x'(t)=f(t,x(t),y(t),x'(t),y'(t)); \quad y''(t)=g(t,x(t),y(t),x'(t),y'(t))$ and the boundary conditions are, for examples, $x(5)=1; ~y(5)=1; ~x(a)=a;~x'(a)y'(a)=2y(a)^2$. Here $a$ is a unknown parameter.
I want to know whether this kind of problem has an official name, and which numerical methods can be used to deal with this kind of problem?
Thanks in advance!!