Exact Differential Equation Condition: ∂M/∂y = ∂N/∂x #195249 (License: Personal Use)

This image shows the classic exactness criterion for a first-order differential equation M(x,y)dx + N(x,y)dy = 0, where equality of mixed partial derivatives (∂M/∂y = ∂N/∂x) ensures the existence of a potential function. The notation is rendered in stylized red and black ink, mimicking lecture notes or textbook marginalia, emphasizing conceptual clarity over formal typesetting.

Used in educational contexts-online math courses, university lecture slides, or STEM blog posts-to illustrate the foundational test for exact differential equations. Matches user intent for learning, reviewing, or teaching ODE theory.