The Murder Mystery Method for Identifying and Solving Exact Differential Equations
Finding out whether a first-order differential equation is exact or not is solving a little "mystery": Is there a function such that its differential change is precisely ? If the answer is "yes", then the equation implies that the change of the function is zero, and therefore the function is equal to a constant,. This last equality is actually an implicit solution of the differential equation .
Tevian Dray and Corinne A. Manogue designed a fun and engaging way to explain how to solve these kind of "mysteries": A crime has been committed (the differential equation), and the student is the detective who must identify the murderer . The detective interrogates (integrates) the witnesses ( and ). If the clues given by them are consistent, then the murderer can be identified (therefore the implicit solution to the differential equation is obtained). If the clues are not consistent, the equation is not exact and has to be solved by another method.
Use the popup menus of this Demonstration to generate different differential equations and see this "Murder Mystery Method" applied to them. Pay attention to the colors and labels of the murderer's clothes and the mathematical expressions, as they show the relationship between the "murder" story and the mathematical procedure.
Tevian Dray and Corinne A. Manogue designed their "Murder Mystery Method" (MMM) in order to determine whether a vector field is conservative or not. The authors of this Demonstration adapted this method to the solution of exact differential equations.
The original authors of the MMM emphasize that "... if two witnesses say they saw someone with red hair, that doesn't mean the suspect has two red hairs! So if you get the same clue more than once, you only count it once..."
T. Dray and C. A. Manogue, "The Murder Mystery Method for Determining Whether a Vector Field Is Conservative," The College Mathematics Journal, 34(3), 2003 pp. 228–231.