Exact Differential Equations

A rigorous method for exact first-order equations, potential functions, integrating checks, and solution domains.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

An equation $M(x,y)\,dx+N(x,y)\,dy=0$ is exact on a region if there exists a potential $F$ with $F_x=M$ and $F_y=N$. Then solutions lie on level curves $F(x,y)=C$. On a simply connected region with continuous first partial derivatives, $M_y=N_x$ is sufficient as well as necessary for exactness.

Notation and mathematical language

Subscripts denote partial derivatives. Integrating $M$ with respect to $x$ gives $F(x,y)=\int M(x,y)\,dx+g(y)$; the 'constant' may depend on $y$. Comparing $F_y$ with $N$ determines $g'(y)$.

Conceptual picture

The differential $dF=F_xdx+F_ydy$ measures change in the potential. Setting $dF=0$ means solution curves remain on a constant-potential contour. Exactness is therefore a conservative-field condition in the plane.

Fully worked example

Interpretation and application

Exact equations model conserved energy and potentials. The constant $C$ labels a trajectory set by initial data. Conservation in the mathematical model does not by itself establish that a physical system has no dissipation; that is a modelling assumption requiring evidence.