I am trying to solve the Hull-White PDE. Let $\theta : \mathbb{R} \to \mathbb{R}$ be a known deterministic function and $\lambda$, $\eta \in \mathbb{R}$. The PDE is then given by
$$f_t(t, r) + \Big(\lambda (\theta(t) - r) \Big) f_r(t, r) + \frac{1}{2} f_{r r}(t, r)\eta^2 = rf(t, r)\text{,}\tag{1}$$
with boundary condition
$$f(T, r) = 1\text{.}\tag{2}$$
I am trying to solve this PDE for times $t<T$, in particular $t=0$.
The typical solution to this problem is to "guess and subsequently verify that the solution has the form"
$$f(t, r) = e^{-rC(t) - A(t)}\text{.}\tag{3}$$
After making this guess, you can plug this solution back into the original PDE and solve for $A(t)$ and $C(t)$.
How would I solve this PDE without guessing and checking? Ideally, the method should be generalizable to PDEs of the form
$$f_t(t, r) + \beta(t, r) f_r(t, r) + \frac{1}{2}(\gamma(t, r))^2f_{r r}(t, r) = rf(t, r)\text{,}\tag{4}$$
where $\beta(t, r)$ and $\gamma(t, r)$ are known and simple.