Exact Differential Equations

An exact differential equation is a first-order differential equation where the left side corresponds directly to the total differential $dF$ of a multi-variable function $F(x, y)$ .

They take the standard form:

M(x, y) \, dx + N(x, y) \, dy = 0

If the expression equals $dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy$ , then the solution to the differential equation is simply the level curves:

F(x, y) = C

The Test for Exactness By Clairaut's theorem on mixed partial derivatives ( $\frac{\partial^2 F}{\partial x \partial y} = \frac{\partial^2 F}{\partial y \partial x}$ ), the equation $M \, dx + N \, dy = 0$ is exact if and only if:

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

Step-by-Step Solution Method

Verify Exactness: Check if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .
Integrate $M$ : Integrate $M(x, y)$ with respect to $x$ , keeping $y$ constant:

F(x, y) = \int M(x, y) \, dx + g(y)

_(where $g(y)$ is an unknown function of $y$ serving as the constant of integration)_

3. Solve for $g(y)$ : Differentiate $F(x, y)$ with respect to $y$ and set it equal to $N(x, y)$ :

\frac{\partial F}{\partial y} = N(x, y) \implies g'(y)

Integrate $g'(y)$ : Find $g(y)$ and substitute it back into $F(x, y)$ .
State the Solution: Set $F(x, y) = C$ .