In elementary ODE textbooks, an early chapter is usually dedicated to first order equations. It is very common to see individual sections dedicated to separable equations, exact equations, and general first order linear equations (solved via an integrating factor), not necessarily in that order.
Common practical applications in these texts include population growth/decay, mixing problems, draining tank/Torricelli's Law problems, projectile motion, Newton's Law of Cooling, orthogonal trajectories, melting snowball type problems, certain basic circuits, growth of an annuity, and logistic population models. (This is just off the top of my head so maybe I am missing other popular ones.) However, all of these end up as separable or first order linear problems and are solved accordingly.
Are there practical applications that lead to first order ODEs which are (exclusively) exact?
Edit: To clarify, I am not saying that exact equations are never useful. I am simply inquiring about their relevance/applicability in the very particular context mentioned above.
To put the question another way, can you briefly state (e.g., in the form of an exercise that would appear in popular undergrad ODE books like Boyce & DePrima; Zill; Nagle/Saff/Snider; Edwards & Penney; etc.) an application problem modeled by a first order exact ODE (which is not separable or linear) and that is "doable" by hand? I've looked in the dozen or so ODE textbooks on my shelf and none of them contain such a problem, and I find that absence curious.