This is for stan-math used from C++; I don’t have the entire stan.

Also, a general thank-you for this very useful AD software.

I noted a difference between the behavior of the cvode bdf and boost rk45 integrator compilation when I switched to the bdf integrator for the first time today.

If I am just integrating a function, where the integrand refers only to the independent variable and the parameters, rather than the current value of the antiderivative, then there is a split between the behavior of these two functions that I wasn’t expecting.

A minimal example follows, with the differences between the two versions called out in the comments:

```
#include "stan/math/mix/mat.hpp"
#include <iostream>
int main(int,char*[]){
stan::math::start_nested();
double t(0.);
std::vector<double> ts{2.};
std::vector<stan::math::var> theta{1.4};
std::vector<double> A0{0.};
std::vector<double> xr;
std::vector<int> xi;
auto a([](auto t,auto&y[[maybe_unused]],auto&th,auto&,auto&,auto*){
return std::vector<decltype(th[0]*t*y[0])>{th[0]*t};});
// A= 2*1.4 = 2.8; next line will fail on line 94 of ode_system.hpp if *y[0] is
// commented out of decltype in the lambda function above due to conversion of
// vector<double> to vector<var>, but won't fail if using integrate_ode_rk45
auto A(stan::math::integrate_ode_bdf(a,A0,t,ts,theta,xr,xi));
A[0][0].grad();
std::cout<<A[0][0]<<", "<<theta[0].adj()<<"\n";//should print: 2.8, 2
stan::math::recover_memory_nested();
return 0;
}
```

Line 94 of ode_system.hpp is in the 2nd Jacobian routine, the one that calculates the Jacobian with respect to both theta and y:

`vector<var> dy_dt_var = f_(t, y_var, theta_, x_, x_int_, msgs_);`

I have no idea if it makes sense to make bdf and rk45 have the same behavior for this use case. It is strange that rk45 is able to compute the same sensitivities even with `*y[0]`

commented out.

For those who might come after, I found that on Windows MinGW GCC, before the cvode version would link, I had to go to `stan-dev/math`

and issue the following command (discovered by reading the makefile therein):

`make CC=g++ lib/cvodes_2.9.0/lib/libsundials_cvodes.a lib/cvodes_2.9.0/lib/libsundials_nvecserial.a`

(and of course add these to LDLIBS in my own project makefile)