How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - \frac{b(T-t)}{2} W^2(t)}$$ Here, W is a Brownian motion.
This is a homework problem, but I missed the class and now struggling with getting it done, could anyone please explain me how to do this one? Thank you