Exact Differential — Definition, Formula & Examples

An exact differential is an expression of the form

M\,dx + N\,dy

that is the total differential of some function

F(x,y)

, meaning

dF = M\,dx + N\,dy

. When a differential equation

M\,dx + N\,dy = 0

has this property, it can be solved by finding

F

directly.

The expression $M(x,y)\,dx + N(x,y)\,dy$ is an exact differential on a simply connected region if and only if $M$ and $N$ have continuous first partial derivatives and satisfy the condition $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ . In that case, there exists a scalar function $F(x,y)$ such that $\dfrac{\partial F}{\partial x} = M$ and $\dfrac{\partial F}{\partial y} = N$ .

Key Formula

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

Where:

$M$ = Function of x and y multiplying dx
$N$ = Function of x and y multiplying dy

How It Works

To solve an exact equation

M\,dx + N\,dy = 0

, first verify exactness by checking

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

. Then integrate

M

with respect to

x

to get

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

, where

g(y)

is an unknown function of

y

alone. Differentiate

F

with respect to

y

and set it equal to

N

to determine

g(y)

. The implicit solution is

F(x,y) = C

Worked Example

Problem: Solve the differential equation (2xy + 3) dx + (x² + 4y) dy = 0.

Test for exactness: Identify M = 2xy + 3 and N = x² + 4y. Compute the partial derivatives.

\frac{\partial M}{\partial y} = 2x, \quad \frac{\partial N}{\partial x} = 2x

Integrate M with respect to x: Since the equation is exact, integrate M to begin finding F(x,y).

F(x,y) = \int (2xy + 3)\,dx = x^2 y + 3x + g(y)

Determine g(y): Differentiate F with respect to y and set equal to N.

\frac{\partial F}{\partial y} = x^2 + g'(y) = x^2 + 4y \implies g'(y) = 4y \implies g(y) = 2y^2

Write the solution: Combine to form the implicit solution.

x^2 y + 3x + 2y^2 = C

Answer: The general solution is

x^2 y + 3x + 2y^2 = C

Why It Matters

Exact differential equations appear throughout physics and engineering whenever a conserved quantity (like energy or potential) exists. Recognizing exactness lets you solve first-order ODEs algebraically instead of relying on more complex techniques. The concept also underpins line-integral path independence in multivariable calculus.

Common Mistakes

Mistake: Forgetting the arbitrary function g(y) when integrating M with respect to x.

Correction: Since you integrate with respect to x only, any pure function of y acts like a constant. Always include g(y) and determine it using the condition ∂F/∂y = N.