Author: John J Weber III, PhDCorresponding 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 $F(x,y)=C$, then $dF=0$ and $\frac{\partial F}{\partial x}}dx+{\displaystyle \frac{\partial F}{\partial y}}dy=0$

or

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

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

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

Definition: Exact Equation

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

Test for Exactness

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