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(x,y)=C$$F(x,y)=C$, then $dF=0$$dF=0$ and ${\displaystyle \frac{\partial F}{\partial x}}dx+{\displaystyle \frac{\partial F}{\partial y}}dy=0$$\dfrac{\partial F}{\partial x}dx+\dfrac{\partial F}{\partial y}dy=0$

or

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

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

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

Definition: Exact Differential Form

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

Definition: Exact Equation

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

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Last Modified: Wednesday, 2 September 2020 07:42 EDT