I know what it means to solve forward/backward difference equations. What I have not been able to wrap my head around is how that formally makes sense in continuous time. To make things concrete, here is an example from an economics paper I have been reading. Take the following differential equation (which describes the evolution of debt $b_t$, I will skip the economics specifics)
$$db_t = s_t dt + r_t b_t dt$$Note that the $t$ subscripts denotes the state of the variable at time $t$. In the paper, the author then "solves forward" this equation to get
$$b_0 = \lim_{T \to \infty} \left[- \int_0^T e^{-\bar{r}_t t}s_tdt + e^{-\bar{r}_T T} b_T\right],$$where $\bar{r}_t = (1/t) \int_0^t r_s ds$. Now, here is my proposed solution. We can write the differential equation (using an integrating factor) as
$$\frac{d}{dt}\left(e^{-\bar{r}_t t} b_t \right) = e^{-\bar{r}_t t}s_t$$
Integrating from $0$ to $T$, we get
\begin{align}\int_0^T \frac{d}{dt}\left(e^{-\bar{r}_t t} b_t \right) dt &= \int_0^T e^{-\bar{r}_t t}s_tdt\\e^{-\bar{r}_T T} b_T - b_0 &= \int_0^T e^{-\bar{r}_t t}s_tdt,\end{align}which, when taking the limit gives us the equation.
So here's my question finally. To me, it seems I just "solved" a differential equation. Where was the "forward" part of my solution, if it is there at all? How would the "backward" solution look, and which part would I need to change?
As is probably clear, I am not well versed in differential equations. So, if possible, try to keep it simple. Thank you!