Final answer:
The questions are related to college-level calculus and involve advanced techniques in solving differential equations and line integrals, suggesting both theoretical and practical mathematical applications.
Step-by-step explanation:
The questions provided appear to be related to advanced mathematics, particularly in the field of calculus and differential equations. The first question involves solving a differential equation that likely requires finding an integrating factor or recognizing if the equation is exact. The second question is looking for the separation of variables in a differential equation with a trigonometric function. These types of problems are typically solved by applying methods like substitution, integration, and separation of variables. However, the excerpts from the text dealing with line integrals, complex functions, Heisenberg's uncertainty principle, and quantum mechanics suggest a broader range of advanced mathematics topics are being discussed overall.
Regarding the example problem, it is a mathematical process where a difficult integral is simplified by changing the variable of integration. In the context of a line integral over a curve, you can choose the variable that makes the integration easier, which might involve converting a complicated function into a simpler form. This example mentions reducing the line integral to an integral over a single variable (either x or y), which is a common strategy in solving these types of integrals. In this particular case, the reduction to variable x is avoided due to the presence of square roots and fractional exponents when expressed in terms of y.