In dynamic programming, the method of undetermined coefficients is sometimes known as "guess and verify." I've periodically heard there are canonical guesses one might make.
In particular, I've seen
$V(k) = A + B\ln(k)$
$V(k) = \frac{Bk^{1-\sigma}}{1-\sigma}$
The former applies to log utility while the latter is related to CRRA preferences. What other canonical guesses exist, and are these generally tied to the particular form of the return function?
Edit: For those not familiar with dynamic programs, what we're trying to do here is come up with closed-forms for the coefficients (e.g. $A$ and $B$). To over-simplify, the functional equation typically takes the generic form $V(k) = \max\bigl\{F(k,u) +\beta V\bigl(g(k,u)\bigr)\bigr\}$, where $g(\cdot,\cdot)$ describes the evolution of the state variable $k$. Essentially, the value of being in state $k$ today depends on today's return function $F(k,u)$ and some discounted value of whatever $k$ is going to be tomorrow $\beta V\bigl(g(k,u)\bigr)$. $u$ represents whatever other non-state variables you think influence the return.
Sometimes it's possible to get a closed-form solution for $V(k)$ (...note: we don't just solve for $V(k)$ since the right-hand side is a maximized quantity). This typically involves knowing something about the return function $F(k,u)$ and then making a guess about the functional form of $V(k)$. We can then iterate to see if our guess yields a closed-form solution for $V(k)$. In particular, this would include closed-forms for the coefficients in the guess (hence the method of undetermined coefficients).