# Exact Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 2.4 – Exact Equations

## Expected Educational Results

• Objective 6–1: I can identify the exact differential form.

• Objective 6–2: I can determine if an ODE is an exact equation.

• Objective 6–3: I can solve exact equations.

## Exact Equations

### Exact Differential Form of an ODE

We will use another notation for total differential in this course. Since $F\left(x,y\right)=C$$F(x,y)=C$, then $dF=0$$dF=0$ and $\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy=0$$\dfrac{\partial F}{\partial x}dx+\dfrac{\partial F}{\partial y}dy=0$

or

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

In reality, the DE will more often be in the form:

$\frac{dy}{dx}=-\frac{M\left(x,y\right)}{N\left(x,y\right)}$$\dfrac{dy}{dx}=-\dfrac{M(x,y)}{N(x,y)}$

#### Definition: Exact Differential Form

The differential form $M\left(x,y\right)\phantom{\rule{0.167em}{0ex}}dx+N\left(x,y\right)\phantom{\rule{0.167em}{0ex}}dy=0$$M(x,y)\,dx+N(x,y)\,dy=0$ is exact in a rectangle $R$$R$ if there is a function $F\left(x,y\right)$$F(x,y)$ such that $\frac{\partial F\left(x,y\right)}{\partial x}=M\left(x,y\right)$$\dfrac{\partial F(x,y)}{\partial x}=M(x,y)$ and $\frac{\partial F\left(x,y\right)}{\partial y}=N\left(x,y\right)$$\dfrac{\partial F(x,y)}{\partial y}=N(x,y)$ for all $\left(x,y\right)$$(x,y)$ in $R$$R$.

#### Definition: Exact Equation

If $M\left(x,y\right)dx+N\left(x,y\right)dy=0$$M(x,y)dx+N(x,y)dy=0$ is in exact differential form, then the DE is called an exact equation.

#### Test for Exactness

Suppose $M\left(x,y\right)$$M(x,y)$ and $N\left(x,y\right)$$N(x,y)$ are continuous in a rectangle $R$$R$, then $M\left(x,y\right)dx+N\left(x,y\right)dy=0$$M(x,y)dx+N(x,y)dy=0$ is an exact equation in $R$$R$ if and only if $\frac{\partial M\left(x,y\right)}{\partial y}=\frac{\partial N\left(x,y\right)}{\partial x}$$\dfrac{\partial M(x,y)}{\partial y}=\dfrac{\partial N(x,y)}{\partial x}$ holds for all $\left(x,y\right)$$(x,y)$ in $R$$R$.