In general, can I find an analytical solution for an equation of the form
$$ \dfrac{dV_{(t)}}{dt} = -\dfrac{V_{(t-1)}}{\tau} $$
where $t$ is time, $V$ is a variable that changes over time, and $\tau$ is a time constant. I think the result should be an exponential decay function, of the form
$$ V_0 \cdot e^{-\tau \cdot t} $$
where $V_0$ is some initial value for $V$.
Also, if anyone could recommend a beginner's book/resource for learning how to solve similar problems I would be most grateful.
Edit: Thanks for LutzL for pointing out this is not an ODE but rather a delay DE.
This is a delay-differential equation (DDE). Usually there is no unique solution, every function on $[−1,0]$ can be extended to a solution over $t≥0$ as $$ V(t)=V(0)-\int_{-1}^{t-1}\frac{V(s)}{τ}\,ds. $$
You can of course also look for solutions of the form $$V(t)=ae^{bt}.$$ Inserting you find $abe^{bt}=-\frac{a}{τ}e^{b(t-1)}\implies bτ=-e^{-b}\iff -\frac1{τ}=be^b$. This has a real solution only if $τ>e$, it can be expressed using the Lambert-W function as $b=W_0(-\frac1{τ})\in[-1,0]$ or $b=W_{-1}(-\frac1{τ})\in(-\infty,-1])$.