Sorry again, for brevity I ignored the initial conditions h as they wont affect the computation speed
B[1:N,1] = rep_vector(1,N);
B[1:N,2] = rep_vector(1,N)+(C[1:N,1].*A[1:N,1].*B[1:N,1]);
Without this initial condition yes there is only one solution, the trivial one of zero